Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
How to add compound fractions? How to add two compound fractions with fractions in numerator like this one:
$$\frac{\ \frac{1}{x}\ }{2} + \frac{\ \frac{2}{3x}\ }{x}$$
or fractions with fractions in denominator like this one:
$$\frac{x}{\ \frac{2}{x}\ } + \frac{\ \frac{1}{x}\ }{x}$$
|
The multiplicative inverse of a fraction a/b is b/a. (Wikipedia)
Let us start with the properties:
*
*Division by a number or fraction is the same as multiplication by its inverse or reciprocal.
Division by $r$ is equal to the multiplication by $\dfrac{1}{r}$:
$$\dfrac{\ \dfrac{p}{q}\ }{r}=\dfrac{p}{q}\cdot \dfr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/51410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Is this Batman equation for real? HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real?
Batman Equation in text form:
\begin{align}
&\left(\left(\frac x7\right)^2\sqrt{\frac{||x|-3|}{|x|-3}}+\left(\frac y3\right)^2\sqrt{\frac{\left|y+\frac{3\sqrt{33}}7\right|}{y+\frac{3\sqrt{... | In fact, the five linear pieces that consist the "head" (corresponding to the third, fourth, and fifth pieces in Shreevatsa's answer) can be expressed in a less complicated manner, like so:
$$y=\frac{\sqrt{\mathrm{sign}(1-|x|)}}{2}\left(3\left(\left|x-\frac12\right|+\left|x+\frac12\right|+6\right)-11\left(\left|x-\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/54506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "466",
"answer_count": 10,
"answer_id": 4
} |
Prove that $\frac4{abcd} \geq \frac a b + \frac bc + \frac cd +\frac d a$
Let $a, b, c$ and $d$ be positive real numbers such that $a+b+c+d = 4$. Prove that $$\frac4{abcd} \geq \frac a b + \frac bc + \frac cd +\frac d a .$$
How can I approach this using only the AM - GM inequality? Are there any other methods that do... | This proof uses the rearrangement inequality in addition to AM-GM. After multiplying by $abcd$ our problem is equivalent to solving $$ a^2cd + ab^2d + abc^2 + bcd^2 \leq 4$$ Let $\{a,b,c,d\}=\{w,x,y,z\}$ with $w \ge x \ge y \ge z$. We have $$ a^2cd + ab^2d + abc^2 + bcd^2 = a(acd)+b(abd)+c(abc)+d(bcd)$$ and by the rear... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/56395",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 2,
"answer_id": 0
} |
$ b_{n + 1} = \frac {b_n^2 + 2b_n}{b_n^2 + 2b_n+2}$ and $ b_1 = 1$, show that $ \left|\frac{2}{n}-\frac{2\ln{n}}{n^2}-b_n\right|\leq\frac{1}{n^2}$ In the recursion $ b_{n + 1} = \frac {b_n^2 + 2b_n}{b_n^2 + 2b_n+2}$, with $ b_1 = 1,$ how can one prove that $ \left|\frac{2}{n}-\frac{2\ln{n}}{n^2}-b_n\right|\leq\frac{1}... | Note that
$$
b_{n+1}+1=2\frac{(b_n+1)^2}{(b_n+1)^2+1}
$$
Letting $b_n=c_n-1$, we get
$$
c_{n+1}=2\frac{c_n^2}{c_n^2+1}
$$
Letting $c_n=\frac{1}{d_n}$, we get
$$
d_{n+1}=\frac{1}{2}(1+d_n^2)
$$
Note that $d_1=\frac{1}{2}$, and if $0\le d_n\le 1$, then $0\le d_{n+1}\le 1$. Thus, $0\le d_n\le 1$ for all $n$.
Letting... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/57376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Does the Schur complement preserve the partial order? Let
$$\begin{bmatrix}
A_{1} &B_1 \\ B_1' &C_1
\end{bmatrix} \quad \text{and} \quad \begin{bmatrix}
A_2 &B_2 \\ B_2' &C_2
\end{bmatrix}$$
be symmetric positive definite and conformably partitioned matrices. If
$$\begin{bmatrix}
A_{1} &B_1 \\ B_1' &C_1
\end{bma... | Yes, it does. The assumption $$\begin{bmatrix}
A_{1} &B_1 \\ B_1^T &C_1
\end{bmatrix}-\begin{bmatrix}
A_2 &B_2 \\ B_2^T &C_2
\end{bmatrix} \geq 0$$ implies that for any vector $\begin{pmatrix} x & y \end{pmatrix}$,
$$ \begin{pmatrix} x^T & y^T \end{pmatrix} \begin{bmatrix}
A_{1} &B_1 \\ B_1^T &C_1
\end{bmatr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/61417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction How can I prove that
$$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$
for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction.
Thanks
| If $S_r= 1^r+2^r+...+n^r=f(n)$ and
$\sigma =n(n+1)$ & $\sigma'=2n+1$
$S_1=\frac{1}{2}\sigma$
$S_2=\frac{1}{6}\sigma\sigma'$
$S_3=\frac{1}{4}\sigma^2$
$S_4=\frac{1}{30}\sigma\sigma'(3\sigma-1)$
$S_5=\frac{1}{12}\sigma^2(2\sigma-1)$
$S_6=\frac{1}{42}\sigma\sigma'(3\sigma^2-3\sigma+1)$
$S_7=\frac{1}{24}\sigma^2(3\sig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/62171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "67",
"answer_count": 16,
"answer_id": 0
} |
Some trigonometric formula How to prove that
$1+2(\cos a)(\cos b)(\cos c)-\cos^2 a-\cos^2 b-\cos^2 c=4 (\sin p)(\sin q) (\sin r)(\sin s)$,
where
$p=\frac{1}{2}(-a+b+c)$, $q=\frac{1}{2}(a-b+c)$, $r=\frac{1}{2}(a+b-c)$, $s=\frac{1}{2}(a+b+c)$.
Thanks.
| Use
$$
\begin{eqnarray}
2 (\sin p)(\sin q) &=& \cos(p-q) - \cos(p+q) \\
2 (\sin r)(\sin s) &=& \cos(r-s) - \cos(r+s) \\
\end{eqnarray}
$$
Then use
$$
\begin{eqnarray}
\cos(p-q) \cos(r-s) &=& \frac{1}{2}( \cos(p+s-q-r) + \cos(p+r - s-q)) \\
\cos(p+q) \cos(r-s) &=& \frac{1}{2}( \cos(p+s+q-r) + \cos(p+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/62349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Complex solutions for Fermat-Catalan conjecture The Fermat-Catalan conjecture is that $a^m + b^n = c^k$ has only a finite number of solutions when $a, b, c$ are positive coprime integers, and $m,n,k$ are positive integers satisfying $\frac{1}{m} + \frac{1}{n} +\frac{1}{k} <1$. There are currently only 10 solutions know... | I found the next complex solution! :)
$$(238+72i)^3+(7+6i)^8=(7347−1240i)^2$$
(There is no new method here. Just small contribution to the problem.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/69291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 2,
"answer_id": 0
} |
Solving a Nonlinear System $$\begin{align}
(1/300)a + (-1/200)b &= 5\\
(-1/300)a + ((-1/300) + (1/200))b + (-1/200)c &= -e^b\\
(-1/200)b + (1/200)c &= -e^c
\end{align}
$$
how do I solve for $a, b$ and $c$?
Thanks!
I know if I derive an equation that isolates a variable, like
$kx + e^x = 0$
I can use Newton's meth... | The first equation is used to substitute all $a \rightarrow \frac{3}{2}b+1500$. The two remaining equations can be collected into a 2x1 vector $f = 0$
$$f = \begin{pmatrix} \hat{e}^b-\frac{b}{300}-\frac{c}{200}-5 \\ \hat{e}^c-\frac{b}{200}+\frac{c}{200} \end{pmatrix} $$
The derivatives with respect to $b$ and $c$ for e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/71446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Is there a combinatorial way to see the link between the beta and gamma functions? The Wikipedia page on the beta function gives a simple formula for it in terms of the gamma function. Using that and the fact that $\Gamma(n+1)=n!$, I can prove the following formula:
$$
\begin{eqnarray*}
\frac{a!b!}{(a+b+1)!} & = & \... | Seven years later I found another way to attack this. Define $f(b, a) = \frac{a!b!}{(a+b+1)!}$ and $h(b, a) = \sum_{i=0}^{b}\binom{b}{i}(-1)^{i}\frac{1}{a+i+1}$. To connect the two, we define $g$ such that $g(0, a) = \frac{1}{a + 1}$ and $g(b + 1, a) = g(b, a) - g(b, a + 1)$ and prove by induction in $b$ that $f = g = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/72067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 2
} |
Partial Fractions of form $\frac{1}{(ax+b)(cx+d)^2}$ When asked to convert something like $\frac{1}{(ax+b)(cx+d)}$ to partial fractions, I can say
$$\frac{1}{(ax+b)(cx+d)} = \frac{A}{ax+b} + \frac{B}{cx+d}$$
Then why can't I split $(cx+d)^2$ into $(cx+d)(cx+d)$ then do
$$\frac{1}{(ax+b)(cx+d)^2} = \frac{A}{ax+b} + \fr... | The degree of the denominator is 2, so the numerator has to be of degree 1. So you can either assume it to be $Bx+C$ or (better still), $B(bx+c)+C$, so that
$\frac{B(bx+c)+C}{(bx+c)^2} = \frac{B}{bx+c} + \frac{c}{(bx+c)^2}.$
This generalizes to the case when the denominator has degree $n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/72281",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
How do I get this matrix in Smith Normal Form? And, is Smith Normal Form unique? As part of a larger problem, I want to compute the Smith Normal Form of $xI-B$ over $\mathbb{Q}[x]$ where
$$
B=\begin{pmatrix} 5 & 2 & -8 & -8 \\ -6 & -3 & 8 & 8 \\ -3 & -1 & 3 & 4 \\ 3 & 1 & -4 & -5\end{pmatrix}.
$$
So I do some elemen... | To expand my comment...Add column 2 to column 1. Subtract row 2 from row 1. Now you have a scalar in the (1,1) position -- rescale to 1.
$$\begin{pmatrix} x-3 & 0 & 0 & 0 \\ 0 & x+1 & 0 & 0 \\ 0 & 0 & x+1 & 0 \\ 0 & 0 & 0 & x+1\end{pmatrix} \sim
\begin{pmatrix} x-3 & 0 & 0 & 0 \\ x+1 & x+1 & 0 & 0 \\ 0 & 0 & x+1 & 0 \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/77063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Prove the identity $ \sum\limits_{s=0}^{\infty}{p+s \choose s}{2p+m \choose 2p+2s} = 2^{m-1} \frac{2p+m}{m}{m+p-1 \choose p}$ $$ \sum\limits_{s=0}^{\infty}{p+s \choose s}{2p+m \choose 2p+2s} = 2^{m-1} \frac{2p+m}{m}{m+p-1 \choose p}$$
Class themes are: Generating functions and formal power series.
| Let $d_s = \binom{p+s}{s} \binom{2p+m}{2p+2s}$. Using the recurrence relations for binomial, the ratio of successive terms is:
$$
\frac{d_{s+1}}{d_s} = \frac{\left(s - m/2\right)\left(s -(m-1)/2\right)}{ (s+1)(s+p+1/2) } = \frac{(s+a)(s+b)}{(s+1)(s+c)}
$$
The hypergeometric certificate above means that
$$
\sum_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/77949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 0
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How do I prove equality of x and y? If $0\leq x,y\leq\frac{\pi}{2}$ and $\cos x +\cos y -\cos(x+y)=\frac{3}{2}$, then how can I prove that $x=y=\frac{\pi}{3}$?
Your help is appreciated.I tried various formulas but nothing is working.
| $cos x +cos y -cos(x+y)=2\cos(\frac{x+y}{2})\cos(\frac{x-y}{2})-(2\cos^2(\frac{x+y}{2})-1)=2\cos(\frac{x+y}{2})\cdot\left(\cos(\frac{x-y}{2})-cos(\frac{x+y}{2})\right)+1\le $
$2\cos(\frac{x+y}{2})\cdot\left(1-cos(\frac{x+y}{2})\right)+1\le 1/2+1=\frac{3}{2}$, where the inequality is by $ab\le (\frac{a+b}{2})^2$, so $\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/78629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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If $2^n+n^2$ is prime number then $n \equiv 0 \pmod 3 $? Is it true that :
$((2^n+n^2) \in \mathbf{P} \land n \geq 3)\Rightarrow n\equiv 0 \pmod 3 $
I have checked this statement for the following consecutive values of $n$ : $3,9,15,21,33,2007,2127,3759$
Note that $2^n+n^2$ is special case of the form $2^n+k\cdot n$... | If $n =1 \mod 6$ then $2^n+n^2 = 0 \mod 3$.
If $n =2 \mod 6$ then $2^n+n^2 = 0 \mod 2$.
If $n =4 \mod 6$ then $2^n+n^2 = 0 \mod 2$.
If $n =5 \mod 6$ then $2^n+n^2 = 0 \mod 3$.
P.S. Looking mod 6 is natural, since $2^n \mod 3$ repeats after 2 steps. So to decide what is $2^n \mod 3$, we need to know if $n$ is odd or eve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/79564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Is the integral $\int_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$ equal for all $a \neq 0$? Let $a$ be a non-zero real number. Is it true that the value of $$\int\limits_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$$ is independent on $a$?
| $$
\begin{align}
I & = \int_0^{\infty} \frac{dx}{(1+x^2)(1+x^a)}\\
\frac{dI}{da} & = -\int_0^{\infty} \frac{x^a \log(x) dx}{(1+x^2)(1+x^a)^2}
\end{align}
$$
Let $\displaystyle J = \int_0^{\infty} \frac{x^a \log(x) dx}{(1+x^2)(1+x^a)^2}$
$$
\begin{align}
J & = \int_0^{\infty} \frac{x^a \log(x) dx}{(1+x^2)(1+x^a)^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/87735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "42",
"answer_count": 4,
"answer_id": 3
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What are some "natural" interpolations of the sequence $\small 0,1,1+2a,1+2a+3a^2,1+2a+3a^2+4a^3,\ldots $? (This is a spin-off of a recent question here)
In fiddling with the answer to that question I came to the set of sequences
$\qquad \small \begin{array} {llll}
A(1)=1,A(2)=1+2a,A(3)=1+2a+3a^2,A(4)=1+2a+3a^2+4a^... | Here is an attack leading to a closed forms for the $A$s and $B$s that can be evaluated for fractional $n$:
First, let $ A_n = \sum_{k=0}^n (k+1)a^k $.
For $n>0$ we can write
$$ A_n = a A_{n-1} + 1 + a + \cdots + a^n = a A_{n-1} + \frac{a^{n+1}-1}{a-1} $$
We can then use a base case of $A_0=1=\frac{a-1}{a-1}$ to unfold... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/88107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
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Completing the square How might I find linear combinations $$\begin{align*}
A&=a_1x+a_2y+a_3z\\
B&=b_1x+b_2y+b_3z\\
C&=c_1x+c_2y+c_3z
\end{align*}$$
Such that I can transform the two polynomials
$$2x^2+3y^2-2yz+3z^2\text{ and }x^2+6xy+3y^2+2yz-6zx+3z^2$$
into
$A^2+B^2+C^2$ and $\alpha A^2+\beta B^2+\gamma C^2$ resp... | Your question is equivalent to finding a matrix $S$ such that $S^\top XS=I$ and $S^\top YS$ is diagonal, where
$$
X=\begin{pmatrix}2&0&0\\ 0&3&-1\\ 0&-1&3\end{pmatrix},
Y=\begin{pmatrix}1&3&-3\\ 3&3&1\\ -3&1&3\end{pmatrix}.
$$
In a related question (BTW, the $X,Y$ here are exactly the same as the matrices $A,B$ in that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/88279",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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$\int \cos^{-1} x \; dx$; trying to salvage an unsuccessful attempt $$
\begin{align}
\int \cos^{-1} x \; dx &= \int \cos^{-1} x \times 1 \; dx
\end{align}
$$
Then, setting
$$\begin{array}{l l}
u=\cos^{-1} x & v=x \\
u' = -\frac{1}{\sqrt{1-x^2}} & v'=1\\
\end{array}$$
Then by the IBP technique, we have:
$$\begin... | So, it looks like you are just having problems with $- \int \frac{x^2}{2} \cdot -x(1-x^2)^{-\frac{3}{2}} \; dx =\frac{1}{2}\int \frac{x^3}{(1-x^2)^{\frac{3}{2}}}dx $.
Lets look at $\int \frac{x^3}{(1-x^2)^\frac{3}{2}}dx$
$$
u=1-x^2 , \text{ then }
$$
$$
du=-2xdx \text{ and }
$$
$$
u-1=x^2
$$
Rewriting,
$$
\b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/88907",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Is this a proper use of induction? ($(n^2+5)n$ is divisible by 6) Just want to get input on my use of induction in this problem:
Question. Use mathematical induction to prove that $(n^2+5)n$ is divisible by $6$ for all integers $n \geqslant 1$.
Proof by mathematical induction.
(1) show base case ($n=1$) is true:
$$
... | $(n^{2} + 5)n \equiv (n^{2}-1)n \equiv (n-1)n(n+1) \pmod 6$. Since these are three consecutive integers, one of them must be congruent to $0\pmod 3$ and one must also be even, or congruent to $0 \pmod 2$. Then as $\gcd (2,3) = 1$, the product $(n-1)n(n+1) \equiv 0 \mod 6$. Altogether meaning that $6 | (n^{2}+5)n$ for a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/90064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Finding any digit (base 10) of a binary number $2^n$ I was doing some Math / CS work, and noticed a pattern in the last few digits of $2^n$.
I was working in Python, in case anyone is wondering.
The last digit is always one of 2, 4, 8, 6; and has a period of 4:
n = 1, str(2 ^ 1)[-1] = 2
n = 2, str(2 ^ 2)[-1] = 4
n = 3,... | The reason for the observer behavior is periodicity of $n \mapsto 2^n \bmod 10$, or $n \mapsto 2^n \bmod 10^k$ for any fixed $k$. This wikipedia article, as well as this, among others, math.SE answer are relevant.
Specifically, for a fixed positive integer $k$, $m \mapsto 2^m \bmod 10^k$ is quasi-periodic with period... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/90926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Planetary Motion: Integral I am solving central force problem to deduce equations of orbit of planets. During the calculation, I am stuck over an integral which I am unable to solve. Can anyone help or guide me in this?
The integral is:
$$\int \dfrac{dr}{r \left(2\mu Er^2 + 2 \mu Cr - l^2 \right)^{1/2}} .$$
Here, $\m... | Make the substitution $\frac1r = x$. Then,
$$
\begin{align*}
\int \frac{dr}{r \left(2\mu Er^2 + 2 \mu Cr - l^2 \right)^{1/2}}
&= \int \frac{dr}{r^2 \left( 2\mu E + \frac{2 \mu C}{r} - \frac{l^2}{r^2} \right)^{1/2}}
\\ &= - \int \frac{dx}{\left( 2\mu E + 2 \mu C x - l^2x^2 \right)^{1/2}}
\\ &= - \int \frac{dx}{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/91923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Deduce that $\mathbb E(X^3)=1^3+2^3+3^3+4^3+5^3+6^3$ A fair die is tossed and let the random variable $X$ be the number that appears.
Deduce that
$$
\mathbb E(X^3)=\frac{1^3+2^3+3^3+4^3+5^3+6^3}6.
$$
First of all, I would like to know the probability distribution of this random variable $X$.
| $X$ takes the values $1$, $2$, $3$ ,$4$, $5$, and $6$ (assuming a six-sided die). Since the die is fair, outcomes are equally likely; so $X$ takes the value $i$ with probability $1/6$ for each $i=1, 2,3,4,5,6$.
The probability distribution of $X$ is therefore
$$
p_X(i)=\textstyle{1\over 6},\quad i=1, 2, \ldots, 6.
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/93407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Need help deriving recurrence relation for even-valued Fibonacci numbers. That would be every third Fibonacci number, e.g. $0, 2, 8, 34, 144, 610, 2584, 10946,...$
Empirically one can check that:
$a(n) = 4a(n-1) + a(n-2)$ where $a(-1) = 2, a(0) = 0$.
If $f(n)$ is $\operatorname{Fibonacci}(n)$ (to make it short), then i... | By inspection $f(3n+3)=4f(3n)+f(3n-3)$, as you’ve already noticed. This is easily verified:
$$\begin{align*}
f(3n+3)&=f(3n+2)+f(3n+1)\\
&=2f(3n+1)+f(3n)\\
&=3f(3n)+2f(3n-1)\\
&=3f(3n)+\big(f(3n)-f(3n-2)\big)+f(3n-1)\\
&=4f(3n)+f(3n-1)-f(3n-2)\\
&=4f(3n)+f(3n-3)\;.
\end{align*}$$
However, I didn’t arrive at this ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/94359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 8,
"answer_id": 2
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Finding the positive integer solutions $(x,y)$ of the equation $x^2+3=y(x+2)$
Finding the positive integer solutions $(x,y)$ of the equation
$x^2+3=y(x+2)$
Source: Art of Problem Solving Vol. 2
Any help would be appreciated.
| Divide the polynomial $x^2+3$ by the polynomial $x+2$. We get
$$\frac{x^2+3}{x+2}=x-2+\frac{7}{x+2}.$$
For an integer $x$, the value of $x-2+\frac{7}{x+2}$ is an integer if and only if $x+2$ divides $7$. The only positive integer $x$ for which this is true is given by $x=5$.
Comment: If we are interested in integer... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Compute: $\int_{0}^{1}\frac{x^4+1}{x^6+1} dx$ I'm trying to compute: $$\int_{0}^{1}\frac{x^4+1}{x^6+1}dx.$$
I tried to change $x^4$ into $t^2$ or $t$, but it didn't work for me.
Any suggestions?
Thanks!
| The denominator of the integrand $f(x):=\dfrac{x^{4}+1}{x^{6}+1}$ may be factored as
\begin{eqnarray*}
x^{6}+1 &=&\left( x^{2}+1\right) \left( x^{4}-x^{2}+1\right) \
&=&\left( x^{2}+1\right) \left( x^{2}-\sqrt{3}x+1\right) \left( x^{2}+\sqrt{3
}x+1\right)
\end{eqnarray*}
If you expand $f(x)$ you get
$$\begin{eqnarray... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/101049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 7,
"answer_id": 0
} |
Simplifying/Converting limits I have this in my lecture:
How did
$$\lim_{x\rightarrow \infty} x^3 \left(\tan{\frac{1}{x}}\right)\left(\sin{\frac{3}{x^2}}\right)$$
become
$$\lim_{x\rightarrow \infty}3\left(\frac{\tan{\frac{1}{x}}}{\frac{1}{x}}\right)\left(\frac{\sin{\frac{3}{x^2}}}{\frac{3}{x^2}}\right)$$
Note the $x... | It is likely a typo. It should probably read:
$$
3 \cdot \frac{\tan(\frac{1}{x})}{\frac{1}{x}} \cdot \frac{\sin(\frac{3}{x^2})}{\frac{3}{x^2}}
$$
which by multiplying the denominator is
$$
\frac{3}{\frac{1}{x} \cdot \frac{3}{x^2}} \cdot \tan\left(\frac{1}{x} \right) \sin\left(\frac{3}{x^2}\right) = x^3 \tan\left(\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/103493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that $x_1^2+x_2^2+x_3^2=1$ yields $ \sum_{i=1}^{3}\frac{x_i}{1+x_i^2} \le \frac{3\sqrt{3}}{4} $ Prove this inequality, if $x_1^2+x_2^2+x_3^2=1$:
$$ \sum_{i=1}^{3}\frac{x_i}{1+x_i^2} \le \frac{3\sqrt{3}}{4} $$
So far I got to $x_1^4+x_2^4+x_3^4\ge\frac{1}3$ by using QM-AM for $(2x_1^2+x_2^2, 2x_2^2+x_3^2, 2x_3^2+x... | Here's a solution with tangent line method.
Note that $$\frac{x}{1+x^2} \le \frac{3\sqrt{3}}{16}(x^2+1) \iff \frac{1}{48}(3x-\sqrt{3})^2(\sqrt{3}x^2+2x+3\sqrt{3}) \ge 0 \text{ (true by discriminant) }$$
Now we have $$\frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2} \le \frac{3\sqrt{3}}{16}(x^2+y^2+z^2+3) = \frac{3\sqrt{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/106473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 4,
"answer_id": 3
} |
Showing $\gcd(n^3 + 1, n^2 + 2) = 1$, $3$, or $9$ Given that n is a positive integer show that $\gcd(n^3 + 1, n^2 + 2) = 1$, $3$, or $9$.
I'm thinking that I should be using the property of gcd that says if a and b are integers then gcd(a,b) = gcd(a+cb,b). So I can do things like decide that $\gcd(n^3 + 1, n^2 + 2) = ... | Playing around along the lines you were exploring should work. Let's do it somewhat casually, aiming always to reduce the maximum power of $n$.
If $m$ divides both $n^3+1$ and $n^2+2$, then $m$ divides $n(n^2+2)-(n^3+1)$, so it divides $2n-1$.
(You got there.)
But if $m$ divides $n^2+2$ and $2n-1$, then $m$ divides $2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 6,
"answer_id": 0
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Direct proof that for a prime $p$ if $p\equiv 1 \bmod 4$ then $l(\sqrt{p})$ is odd.
Definition: Assume $p$ is a prime. $l(\sqrt{p})=$ length of period in simple continued fraction expansion of $\sqrt{p}$.
The standard proof of this uses the following:
*
*$p$ is a prime implies $p \equiv 1 \bmod 4$ iff $x^2-py^2=-1... | We can deduce both of (1) and (2), and the fact that $p$ is a sum of two squares, by playing with quadratic forms. I have the impression this is all "well known to those who know it well". But I've never seen this elementary argument written down in elementary language, so here it is.
We begin with some general discuss... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 1,
"answer_id": 0
} |
A tricky double integral What is
$$\int_0^1 \int_0^1 \frac{ dx \; dy}{1+xy+x^2y^2} ? $$
Can you do one of the integrals and turn it into a single integral?
I get lost in a sea of inverse tangents.
| The first step would be one $\arctan$ only.
The denominator is
$$1+xy+x^2y^2 =\left(\frac{1}{2}+xy\right)^2 + \frac{3}{4}$$
Hence, with $a=\frac{2}{\sqrt{3}}$ we have
$$\int\frac{dy}{1+xy+x^2y^2} =\int\frac{dy}{\left(\frac{1}{2}+xy\right)^2 + 1/a^2}=\qquad\qquad\qquad\text{ }\\
a^2\int\frac{dy}{a^2\lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/111319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Show that $\tan 3x =\frac{ \sin x + \sin 3x+ \sin 5x }{\cos x + \cos 3x + \cos 5x}$ I was able to prove this but it is too messy and very long. Is there a better way of proving the identity? Thanks.
| The identities for the sum of sines and the sum of cosines yield
$$
\frac{\sin(x)+\sin(y)}{\cos(x)+\cos(y)}=\frac{2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)}{2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)}=\tan\left(\frac{x+y}{2}\right)\tag{1}
$$
Equation $(1)$ implies that
$$
\frac{\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/113451",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 6,
"answer_id": 0
} |
Integers and fractions How would I write this as an integer or a fraction in lowest terms?
$(1-\frac12)(1+\frac 12)(1-\frac13)(1+\frac13)(1-\frac14)(1+\frac14).....(1-\frac1{99})(1+\frac1{99})$
I really need to understand where to start and the process if anyone can help me.
| Hint: telescopy, note how the adjacent like-colored terms all cancel out of the products below
$$\rm \left(1-\frac{1}2\right)\left(1-\frac{1}3\right)\cdots \left(1-\frac{1}n\right)\ =\ \frac{1}{\color{red}2} \frac{\color{red}2}{\color{green}3} \frac{\color{green}3}{\color{blue}4} \frac{\color{blue} 4}{}\: \cdots\: \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/114211",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Bennett's Inequality to Bernstein's Inequality Bennett's Inequality is stated with a rather unintuitive function,
$$
h(u) = (1+u) \log(1+u) - u
$$
See here. I have seen in multiple places that Bernstein's Inequality, while slightly weaker, can be obtained by bounding $h(u)$ from below,
$$
h(u) \ge \frac{ u^2 }{ 2 + \f... | While the answer by @Arash outlines the general strategy of deriving the inequality,
the explicit computation is missing.
I am providing it here in the hope that it will be helpful for future readers.
The actual computation of the derivative is highly simplified by first simplifying
the last term in the definition of
$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 3
} |
Ratio test for $\frac{\sqrt{n^n}}{2^n}$ Before I tried root test, I did ratio test for $\frac{\sqrt{n^n}}{2^n}$ and got:
$$\lim_{n\to\infty}\frac{\sqrt{(n+1)^{(n+1)}}}{2^{n+1}}\cdot \frac{2^n}{\sqrt{n^n}} = \lim_{n\to\infty} \frac{1}{2}\cdot \sqrt{\frac{(n+1)^{(n+1)}}{n^n}} = \frac{1}{2}$$
But correct answer with rati... | The ratio of consecutive terms is indeed $$\frac{1}{2}\cdot \sqrt{\frac{(n+1)^{(n+1)}}{n^n}}$$ but that second factor does not tend to $1$ like you seem to have assumed. We actually have $$ \sqrt{\frac{(n+1)^{(n+1)}}{n^n}} = \sqrt{n+1} \cdot \sqrt{ \left( 1+ \frac{1}{n} \right)^n } .$$
Since $\left( 1+ \frac{1}{n} \rig... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Evaluating $\lim_{x \to 0}\frac{1-\cos (1- \cos x)}{x^4}$ $$\lim_{x \to 0}\frac{1-\cos (1- \cos x)}{x^4}$$
I don't think L'Hôpital's rule is a good idea here.
I will not finish this until the evening and it's easy to make mistake. Maybe I can expand $\cos$ in a series?
But I don't know how to use this trick...
| You can expand cos in a series, like you said:
$$1 - \cos\left(1 - \cos x\right) = 1 - \cos\left(1 - \left(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\right)\right) $$
$$= 1 - \cos\left(\frac{x^2}{2!} - \frac{x^4}{4!} + \cdots\right) $$
$$= 1 - \left[1 - \frac{1}{2!}\left(\frac{x^2}{2!} - \frac{x^4}{4!} + \cdots\right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/118871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Trigonometric identity, possible error I need to prove the following trigonometric identity:
$$ \frac{\sin^2(\frac{5\pi}{6} - \alpha )}{\cos^2(\alpha - 4\pi)} - \cot^2(\alpha - 11\pi)\sin^2(-\alpha - \frac{13\pi}{2}) =\sin^2(\alpha)$$
I cannot express $\sin(\frac{5\pi}{6}-\alpha)$ as a function of $\alpha$. Could it b... | Some important translations:
$$\tag 1\sin(x\pm 2 \pi) = \sin x $$
$$\tag {1'}\cos(x\pm 2 \pi) = \cos x $$
$$\tag 2\cot(x\pm \pi)= \cot x$$
$$\tag {2'}\tan(x\pm \pi)= \tan x$$
$$\tag 3 \sin \left(\frac \pi 2 -x \right)=\cos x$$
$$\tag 4 \cos \left(\frac \pi 2 -x \right)=\sin x$$
$$\tag 5 \sin(\pi-x)=\sin x$$ and $$\tag... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/119481",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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How to integrate $\int_0^\infty\frac{x^{1/3}dx}{1+x^2}$? I'm trying to evaluate the integral $\displaystyle\int_0^\infty\frac{x^{1/3}dx}{1+x^2}$.
My book explains that to evaluate integrals of the form $\displaystyle\int_0^\infty x^\alpha R(x)dx$, with real $\alpha\in(0,1)$ and $R(x)$ a rational function, one first sta... | Suppose $\alpha \in (-1, 1)$. Consider a keyhole contour $\Gamma$ of outer radius $R$ and inner radius $\varepsilon$ about the positive real axis. Let $\gamma_R$ denote the outer arc, $\gamma_\varepsilon$ denote the inner arc, and $\gamma_\pm$ denote the segment going away from and towards the origin. Note that this co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/121976",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 3
} |
Find the largest positive value of $x$ at which the curve $y = (2x + 7)^6 (x - 2)^5$ has a horizontal tangent line. I need help with the following question.
Find the largest positive value of x at which the curve:
$$y = (2x + 7)^6 (x - 2)^5$$
has a horizontal tangent line.
| Hint: $\displaystyle{\frac{dy}{dx} = \left(12(2x+7)^5(x-2)^5+5(2x+7)^6(x-2)^4 \right) = 0}$ , in other words
$\displaystyle{\frac{dy}{dx} = (2x+7)^5(x-2)^4(22x+11) = 0}$ at what points?
$\displaystyle{\frac{dy}{dx} = 0}$ at $x=-\frac{7}{2}, x=2, x=-\frac{1}{2}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/122226",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Laurent Series of rational function about $z=0$ Find the Laurent expansion for $$\frac{\exp(\frac 1{z^2})}{z-1}$$ about $z=0$. I know that $\exp( \frac 1{z^2}) = \sum_{n=0}^\infty \frac{z^{-2n}}{n!}$
and $ \frac 1{z-1}=-\sum_{n=0}^\infty z^n$
| Write $$\exp(1/z^2) = \sum_{n=-\infty}^\infty a_n z^n \qquad\text{and}\qquad \frac{1}{z-1} = \sum_{n=-\infty}^\infty b_n z^n,$$
i.e. $a_{-2n} = 1/n!$ and $a_k = 0$ otherwise, and $b_n = -1$ for $n \ge 0$ and $b_n = 0$ for $n < 0$. Multiply the two series:
$$ \left(\sum_{n=-\infty}^\infty a_n z^n\right)\left(\sum_{n=-\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/122368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Exponential Equation with mistaken result I'm on my math book studying exponential equations, and I got stuck on this Problem:
What is sum of the roots of the equation:
$$\frac{16^x + 64}{5} = 4^x + 4$$
I decided to changed: $4^x$ by $m$, so I got: $$\frac{m^2 + 64}{5} = m + 4$$
working on it I've got: $m^2 - 5m + 44 ... | Perhaps you copied the equation wrong. For the equation
$$ \frac{16^x + 64}{5} = 4^x + b$$
(where presumably $x$ is supposed to be real), substituting $m = 4^x$ we get
$$m = \frac{5 \pm \sqrt{20 b - 231}}{2}$$
where we want both solutions to be real, so $11.55 \le b < 12.8$. Now
$x_1 + x_2 = k$ where $m_1 m_2 = 4^{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/124456",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Binomial Coefficients in the Binomial Theorem - Why Does It Work Question to keep it simple: Given
$(a+b)^3=\binom{3}{0}a^3+\binom{3}{1}a^2b+\binom{3}{2}ab^2+\binom{3}{3}b^3$
Could you please give me an intuitive combinatoric reason to why the binomial coefficients are here?
for instance, what does $\binom{3}{2}ab^2$ m... | This isn't as rigorous as the accepted answer, but explores the question a bit from another perspective.
Take your example,
$(a+b)^3=\binom{3}{0}a^3+\binom{3}{1}a^2b+\binom{3}{2}ab^2+\binom{3}{3}b^3$
Let us consider this small rewrite of the same example:
$(a+b)^3=\binom{3}{0}{a}\cdot{a}\cdot{a}+
\binom{3}{1}{a}\cdot{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/127926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
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Square Root Of A Square Root Of A Square Root Is there some way to determine how many times one must root a number and its subsequent roots until it is equal to the square root of two or of the root of a number less than two?
sqrt(16)=4
sqrt(4)=2
sqrt(2) ... 3
--
sqrt(27)=5.19615...
sqrt(5.19615...)=2.27950...
sqrt(2.2... | Taking the square root $n$ times is taking the $2^n$-th root. If the $2^n$-th root of $x$ is $\le\sqrt2$, but the $2^{n-1}$-st root of $x$ is $>\sqrt2$, then
$$x^{1/2^n}\le \sqrt2<x^{1/2^{n-1}}\;,$$
which, after raising everything to the $2^n$ power, is equivalent to
$$x\le (2^{1/2})^{2^n}<x^2\;,$$
or $$x\le 2^{2^{n-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/129446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
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Number of positive integral solutions of equation $\dfrac 1 x+ \dfrac 1 y= \dfrac 1 {n!}$ $$\dfrac 1 x+ \dfrac 1 y= \dfrac 1 {n!}$$
This is one of the popular equation to find out the number of solutions. From Google, here I found that for equation $\dfrac 1 x+ \dfrac 1 y= \dfrac 1 {n!}$, number of solutions are
$\psi... | If we let $N! = M$,
$$ \frac{1}{x}+\frac{1}{y} = \frac{1}{N!} = \frac{1}{M}$$
Let $x=M+a$ and $y=M+b$ where $a$ and $b$ are integers (positive or negative)
$$ \Rightarrow \frac{2M+a+b}{(M+a)(M+b)} = \frac{1}{M}$$
$$ 2M^2+M(a+b) = M^2+M(a+b)+ab$$
$$ \Rightarrow M^2 = ab$$
Now look at all the divisors of $M^2 = (N!)^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/131050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 2,
"answer_id": 1
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Fastest increase of a function For the function $f(x,y,z) = \frac{1 }{ x^2+y^2+z^2}$ what is the direction of the fastest increase at $(1,1,1)$?
| In the direction of the gradient of $f$ evaluated at $(1,1,1)$.
This is a general fact concerning functions $f=f(x,y,z)$: The direction in which a differentiable function $f$ increases most rapidly at the point $(a,b,c)$ is in the direction of $\nabla f(a,b,c)$.
You'll need to find the general formula for $\nabla f... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is $\lim\limits_{k \to 0}{f(k) = 2 + k^{\frac{3}{2}}\cos {\frac{1}{k^2}}}$ Just want to check this one:
I got:
$$\displaystyle \lim_{k \to 0}{f(k) = 2} \;+\; \lim_{k \to 0}{k^{\frac{3}{2}}\cos {\frac{1}{k^2}}}$$
Since $\lim\limits_{k \to 0}\cos{\frac{1}{k^2}} = 0$, using the squeeze theorem, I have $\lim\limits_{k... | Almost. Since $\lim\limits_{k\rightarrow 0^+} k^{3/2}=0$ (note the one-sided limit) and since $-1\le \cos(x)\le1$ for all $x$, it follows from the Squeeze Theorem that $\lim\limits_{k\rightarrow 0^+} \bigl[\,k^{3/2}\cos(1/k^2)\,\bigr]=0$.
Thus, $\lim\limits_{k\rightarrow 0^+} \bigl[2+k^{3/2}\cos(1/k^2)\,\bigr]=2+0=2$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/135614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to calculate $\int{\frac{dx}{3x^2+2}}$? I've started doing $$\displaystyle\int{\dfrac{dx}{3x^2+2}}$$
but I only get
$$\displaystyle\int{(3x^2+2)^{-1}dx}\\
\frac{1}{6}\displaystyle\int{\frac{6x(3x^2+2)^{-1}}{x}dx}\\
$$
And I don't know how to do solve this.
| If we put $\sqrt{\frac{3}{2}}x=t$ then it means that $\mathrm dx=\sqrt{\frac{2}{3}}\mathrm dt$
$$\begin{eqnarray*}
\int{\dfrac{\mathrm dx}{3x^2+2}} &=& \displaystyle \frac{1}{2}\int{\dfrac{\mathrm dx}{(\sqrt{\frac{3}{2}}x)^2+1}}\\
&=& \displaystyle \frac{1}{2}\sqrt{\frac{2}{3}}\int{\dfrac{\mathrm dt}{t^2+1}}\\
&=& \dis... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/137765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
How can I show using mathematical induction that $\frac{1}{2} + \frac{1}{4} + \cdots + \frac{1}{2^n} = \frac{2^n - 1}{2^n}$ How can I show using mathematical induction that $\frac{1}{2} + \frac{1}{4} + \cdots + \frac{1}{2^n} = \frac{2^n - 1}{2^n}$
Edit: I'm specifically stuck on showing that $\frac{2^n - 1}{2^n} + \fra... | To prove that
$$\frac{1}{2} + \frac{1}{4} + \cdots + \frac{1}{2^n} = \sum_{k=1}^n \frac{1}{2^k} = \frac{2^n - 1}{2^n}$$
We can use two steps with telescopic induction.
This form of induction works as follows: To prove that
$$\sum_{k=1}^n f(k)=g(n)$$
Induction can be used to prove this equality by confirming the follo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/141126",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 4
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If $x$ and $y$ are rational numbers and $x^5+y^5=2x^2y^2,$ then $1-xy$ is a perfect square. Prove that if $x, y$ are rational numbers and
$$ x^5 +y^5 = 2x^2y^2$$
then $1-xy$ is a perfect square.
| I would be so tempted to divide by $x^2 y^2$, so I would consider the following cases:
Case a:
if $x=0$ then that implies $y=0$.
Case b:
if $y=0$ then that would imply $x=0$
Case c:
$x \neq 0, y \neq 0$
Thus dividing by $x^2y^2$ will be legal
$$
\begin{align*}
\frac{x^3}{y^2} - 2 + \frac{y^3}{x^2} = 0 \\
x\left(\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/141475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 4,
"answer_id": 3
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Show that $\forall n \in \mathbb{N} \left ( \left [(2+i)^n + (2-i)^n \right ]\in \mathbb{R} \right )$ Show that $\forall n \in \mathbb{N} \left ( \left [(2+i)^n + (2-i)^n \right ]\in \mathbb{R} \right )$
My Trig is really rusty and weak so I don't understand the given answer:
$(2+i)^n + (2-i)^n $
$= \left ( \sqrt{5} ... | There are two ways to write a complex number: rectangular form, e.g., $x+iy$, and polar form, e.g., $re^{i\theta}$. The conversion between them uses trig functions: $$re^{i\theta}=r\cos\theta+ir\sin\theta\;.\tag{1}$$ Going in the other direction, $$x+iy=\sqrt{x^2+y^2}\,e^{i\theta}\;,$$ where $\theta$ is any angle such ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/144901",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Calculating $\tan15\cdot \tan 30 \cdot \tan 45 \cdot \tan 60 \cdot \tan 75$ What is $\tan15\cdot \tan 30 \cdot \tan 45 \cdot \tan 60 \cdot \tan 75$ equal to (in degrees)?
Here is how I tried to solve it:
I assumed $x = \tan 15 (x= 0.27)$, so I rewrote it as:
$x\cdot 2x \cdot 3x \cdot 4x \cdot 5x = 120x^5$
$0.27^5 = 0... | EDIT : Arturo's solution is really cool. Use that instead, I'll let my solution stay just as a reference.
$$\tan(2A) \neq 2\times \tan(A)$$
You might want to read/revise double angle formulas.
Assuming you need to do this without a calculator,
$$\begin{align*}
\tan 30 &= \frac{1}{\sqrt{3}}\\
\tan 45 &= 1\\
\tan 60 &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/146808",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Integrating $ \int_0^{2\pi} \frac{5}{2}|\sin(2t)| dt$ I'm stuck on integration of the following function:
$$ \int_0^{2\pi} \frac{5}{2}|\sin(2t)| dt$$
I understand the few first steps with substitution, etc., but I can't get the end result which is "10".
Could someone give me a step by step solution for it?
| We have
$$\begin{equation}
\int^{t=2 \pi}_{t=0} \frac{5}{2} |\text{sin} 2t | \, dt = \frac{5}{2} \int^{t=2 \pi}_{t=0} |\text{sin} 2t | \, dt
\end{equation}$$
Now use the substitution $y = 2t$, $dy = 2dt$. The boundaries shift as follows: if $t = 2\pi$ then $y = 4\pi$ and if $t = 0$ then $y = 0$
We plug this into the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/147495",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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If $n\equiv 2\pmod 3$, then $7\mid 2^n+3$. In this (btw, nice) answer to Twin primes of form $2^n+3$ and $2^n+5$, it was said that:
If $n\equiv 2\pmod 3$, then $7\mid 2^n+3$?
I'm not familiar with these kind of calculations, so I'd like to see, if my answer is correct:
*
*Let $n=3k+2$ so then
$2^{3k+2}+3\equiv 2^{... | In fact, we can prove a stronger result and the proof is easier. The result we will prove is that $$x^2+x+1 \text{ divides }x^{3k+2} + x+1$$ for all $k \in \mathbb{N}$. Setting $x=2$ gives the result, you are looking for.
The proof follows immediately from the remainder theorem since $(x^2+x+1) = (x-\omega)(x-\omega^2)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/148418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Folding a rectangular paper sheet You are given a rectangular paper sheet. The diagonal vertices of the sheet are brought together and folded so that a line (mark) is formed on the sheet. If this mark length is same as the length of the sheet, what is the ratio of length to breadth of the sheet?
This is my first questi... | $\hskip 2.2in$
The above figure was done using grapher on mac osx.
Let $l$ be the length (i.e. the sides $AD$ and $BC$) and $b$ be the breadth (i.e. the sides $AB$ and $CD$). Once you get the diagonal vertices together, i.e. when $D$ coincides with $B$, the length $EB$ is the same as the length $ED = l-a$.
Hence, for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/148612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Is $k=3$ the only solution to $\sum_{r=1}^n r^k =\left( \sum_{r=1}^nr^1 \right)^2 $? $f(k) = \sum_{r=1}^{n} r^k$. Find an integer $x$ that solves the equation $f(x) = \bigl(f(1)\bigr)^2$.
Problem credit: http://cotpi.com/p/2/
I understand why $x = 3$ is a solution. $1^3 + 2^3 + \dots + n^3 = \left(\frac{n(n + 1)}{2}\ri... | I assume that you want to find values of $x$ for which $f(x) = (f(1))^2$ as functions in $n$. In this case, we can simply substitute in $n=1$, $n=2$, and $n=3$. Namely, we need to satisfy $$1^x=1 \qquad \text{and} \qquad 1^x + 2^x = 9 \qquad \text{and} \qquad 1^x + 2^x + 3^x = 36$$ Subtracting successive equations yiel... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/149223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
How to show that $n(n^2 + 8)$ is a multiple of 3 where $n\geq 1 $ using induction? I am attempting a question, where I have to show $n(n^2 + 8)$ is a multiple of 3 where $n\geq 1 $.
I have managed to solve the base case, which gives 9, which is a multiple of 3.
From here on,
I have $(n+1)((n+1)^2 + 8)$
$n^3 + 3n^2 + 1... | Since you have proven that this formula is available for $n=1$ we suppose that formula is available for all natural numbers n=k
$$k(k^2+8)$$ then according to axiomme of mathematical induction we need to prove that formula is valid for $n=k+1$ or
$$(k +1)((k+1)^2+8)$$ is multiple of 3
now we have
$$(k+1)((k+1)^2+8)=(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/150425",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 9,
"answer_id": 7
} |
Show that $11^{n+1}+12^{2n-1}$ is divisible by $133$. Problem taken from a paper on mathematical induction by Gerardo Con Diaz. Although it doesn't look like anything special, I have spent a considerable amount of time trying to crack this, with no luck. Most likely, due to the late hour, I am missing something very tr... | In fact, we can prove a stronger result and the proof is easier. The result we will prove is that $$x^2+x+1 \text{ divides }x^{n+1} + (x+1)^{2n-1}$$ for all $n \in \mathbb{N}$. Setting $x=11$ gives the result, you are looking for.
The proof follows immediately from the remainder theorem since $(x^2+x+1) = (x-\omega)(x-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/150979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 5,
"answer_id": 4
} |
Real roots of $3^{x} + 4^{x} = 5^{x}$ How do I show that $3^{x}+4^{x} = 5^{x}$ has exactly one real root.
| Rewrite our equation as
$$\left(\frac{3}{5}\right)^x+\left(\frac{4}{5}\right)^x=1.$$
We have the familiar solution $x=2$.
If $x>2$, then $\left(\frac{3}{5}\right)^x \lt \left(\frac{3}{5}\right)^2$ and
$\left(\frac{4}{5}\right)^x \lt \left(\frac{4}{5}\right)^2$, and therefore
$$\left(\frac{3}{5}\right)^x+\left(\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/153614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Long division in integration by partial fractions I am trying to figure out what my book did, I can't make sense of the example.
"Since the degree of the numberator is greater than the degree of the denominator, we first perform the long division. This enables us to write
$$\int \frac{x^3 + x}{x -1} dx = \int \left(x^2... | You want to express $x^3+x$ as $(x-1)(\text{something})+r$.
I want to show you a way of solving this problem with a homemade technique. We see that the "something" must be a polynomial of degree $2$, or either we'll be getting $x^4$ which we don't want.
$$x^3+x=(x-1)(ax^2+bx+c)+r$$
If we multiply out we get
$$x^3+x=a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/154008",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $\sin(2A)+\sin(2B)+\sin(2C)=4\sin(A)\sin(B)\sin(C)$ when $A,B,C$ are angles of a triangle
Prove that $\sin(2A)+\sin(2B)+\sin(2C)=4\sin(A)\sin(B)\sin(C)$ when $A,B,C$ are angles of a triangle
This question came up in a miscellaneous problem set I have been working on to refresh my memory on several topics I... | Since $A,B,C$ are angles of a triangle, we have that $A+B+C = \pi$. Recall the following trigonometric identities
\begin{align}
\sin(\pi-\theta) & = \sin(\theta)\\
\cos(\pi-\theta) & = -\cos(\theta)\\
\sin(2\theta) + \sin(2\phi) & = 2 \sin(\theta + \phi) \cos(\theta-\phi)\\
\sin(2\theta) & = 2\sin(\theta) \cos(\theta)\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/154505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 3,
"answer_id": 1
} |
Polar equation of a circle A very long time ago in algebra/trig class we did polar equation of a circle where
$r = 2a\cos\theta + 2b\sin\theta$
Now I forgot how to derive this. So I tried using the standard form of a circle.
$$(x-a)^2 + (y - b)^2 = a^2 + b^2$$
$$(a\cos\theta - a)^2 + (b\sin\theta - b)^2 = a^2 + b^2$$
$... | Although it's already been answered, it seemed like a fun way to procrastinate on my homework.
$(x-a)(x-a) + (y-b)(y-b) = aa + bb$
Substitute $x = rcos(θ)$ and $y = rsin(θ)$.
$(rcos(θ) - a)(rcos(θ) - a) + (rsin(θ) - b)(rsin(θ) - b) = aa + bb$
Multiply using FOIL.
$rrcos(θ)^2 - 2arcos(θ) + aa + rrsin(θ)^2 - 2brsin(θ) + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/154550",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 4
} |
How to solve this quartic equation? For the quartic equation:
$$x^4 - x^3 + 4x^2 + 3x + 5 = 0$$
I tried Ferrari so far and a few others but I just can't get its complex solutions. I know it has no real solutions.
| Let $f(x) = x^4 - x^3 +4x^2 +3x+5$. Once you know that $f(x)$ doesn't have real solutions, try some easy complex numbers like $i$, $\omega$, other $n^{th}$ roots of unity etc. Note that if we plug in $x = \omega$, where $\omega$ is the complex cube-root of unit, we get that $$f(\omega) = \omega^4 - \omega^3 +4 \omega^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155242",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
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Simplify these expressions with radical sign 2 My question is
1) Rationalize the denominator:
$$\frac{1}{\sqrt{2}+\sqrt{3}+\sqrt{5}}$$
My answer is:
$$\frac{\sqrt{12}+\sqrt{18}-\sqrt{30}}{18}$$
My question is
2) $$\frac{1}{\sqrt{2}+\sqrt{3}-\sqrt{5}}+\frac{1}{\sqrt{2}-\sqrt{3}-\sqrt{5}}$$
My answer is: $$\frac... | *
*Your answer is almost correct.
Multiplying by$$\frac{\sqrt{2}+\sqrt{3}-\sqrt{5}}{\sqrt{2}+\sqrt{3}-\sqrt{5}}$$ and simplifying will give your that answer:
$$\frac{\sqrt{2}+\sqrt{3}-\sqrt{5}}{2 \sqrt 6}=\frac{\sqrt{12}+\sqrt{18}-\sqrt{30}}{12}$$
2. Your answer is correct.
Multiplying the first fraction by $$\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Find the square root of the polynomial My question is:
Find the square root of the polynomial-
$$\frac{x^2}{y^2} + \frac{y^2}{x^2} - 2\left(\frac{x}y + \frac{y}x\right) + 3$$
| $$\frac{x^2}{y^2} + \frac{y^2}{x^2} - 2\left(\frac{x}y + \frac{y}x\right) + 3$$
$$=\frac{{x^4} + {y^4} - 2\left({x^2} +{y^2}\right)xy + 3{x^2}{y^2}}{{x^2}{y^2}}$$ $$=\frac{{x^4} + {y^4} + 2{x^2}{y^2}+ {x^2}{y^2} - 2\left({x^2} +{y^2}\right)xy }{{x^2}{y^2}}$$ $$=\frac{({x^2}+{y^2})^2+ {(xy)^2} - 2\left({x^2} +{y^2}\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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Leslie Matrix characteristic polynomial I´m having problems to prove the Leslie matrix characteristic polynomial.
I have to prove that the characteristic polynomial is:
$$
\ λ^{n}-a_{1}λ^{n-1}-a_{2}b_{1}λ^{n-2}-a_{3}b_{1} b_{2}λ^{n-3} - ... -
a_{n}b_{1} b_{2}...b_{n-1}\
$$
I would apreciate some light!
| I'm assuming (from Wikipedia) that the matrix is $$\begin{pmatrix} a_1 & a_2 & a_3 & a_4 & ... & a_n \\ b_1 & 0 & 0 & 0 & ... & 0 \\ 0 & b_2 & 0 & 0 & ... & 0 \\0 & 0 & b_3 & 0 & ... & 0 \\ 0 & 0 & 0 & .. & .. & 0 \\ 0 & 0 & 0 & ... & b_{n-1} & 0\end{pmatrix}$$
Use induction on $n$. For $n = 1$ the proof is easy.
For ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/158877",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Euler's product formula for $\sin(\pi z)$ and the gamma function I want to derive Euler's infinite product formula
$$\displaystyle \sin(\pi z) = \pi z \prod_{k=1}^\infty \left( 1 - \frac{z^2}{k^2} \right)$$
by using Euler's reflection equation $\Gamma(z)\Gamma(1-z) \sin(\pi z) = \pi$ and the definition of $\Gamma(z)$ a... | Use the fact that $\Gamma (1-z) = -z\, \Gamma(-z)$ and then:
$$\Gamma(1-z)\Gamma(z) = -z \, \Gamma(-z)\Gamma(z) = -z \cdot \frac{1}{-z}\cdot \frac{1}{z} \prod_{k=1}^{+\infty} \frac{1}{\left(1 + \frac{z}{k} \right)\left(1 - \frac{z}{k} \right) } = \frac{1}{z} \prod_{k=1}^{+\infty} \frac{1}{1 - \frac{z^2}{k^2}} = \frac{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/159862",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
A simple quadratic inequality For positive integers $n\ge c\ge 5$, why does
$$c+2(n-c)+\frac{(n-c)^2}{4}\le\frac{(n-1)^2}{4}+1\text{ ?}$$
| To avoid fractions, we multiply the left-hand side by $4$, obtaining
$$(n-c)^2+8(n-c)+4c.$$
Complete the square. We get
$$(n-c+4)^2 +4c -16.$$
Now calculate $[(n-1)^2 +4]-[(n-c+4)^2 +4c -16]$.
The difference of squares factors as $(c-5)(2n-c+3)$, so
$$\begin{align}[(n-1)^2 +4]-[(n-c+4)^2 +4c -16]&=(c-5)(2n-c+3)-4(c-5... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/159997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Linear algebra: power of diagonal matrix? Let A = $\begin{pmatrix} 3 & -5 \\ 1 & -3 \end{pmatrix}$. Compute $A^{9}$. (Hint: Find a matrix P such that $P^{-1}AP$ is a diagonal matrix D and show that $A^{9}$= $PD^{9}P^{-1}$
Answer: $\begin{pmatrix} 768 & -1280 \\ 256 & -768 \end{pmatrix}$
I keep getting $\begin{pmatrix} ... | Your answer is not correct. Please note that the eigenvectors should be corresponding to the eigenvalues. So, if you choose $$D=\left(
\begin{array}{cc}
-2 & 0 \\
0& 2
\end{array}
\right),$$ then your $P$ should be $$P=\left(
\begin{array}{cc}
1 & 5 \\
1& 1
\end{array}
\right),$$
because $(1,1)$ is the eigenvector ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/162516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How to find the sum of this infinite series. How to find the sum of the following series ?
Kindly guide me about the general term, then I can give a shot at summing it up.
$$1 - \frac{1}{4} + \frac{1}{6} -\frac{1}{9} +\frac{1}{11} - \frac{1}{14} + \cdots$$
| Using the principal value for the doubly infinite harmonic series yields
$$
\begin{align}
\sum_{k=0}^{\infty} \left(\dfrac1{5k+1} - \dfrac1{5k+4}\right)
&=\frac15\sum_{k=-\infty}^\infty\frac{1}{k+1/5}\\
&=\frac{\pi}{5}\cot\left(\frac{\pi}{5}\right)\\
&=\frac{\pi}{5}\sqrt{\frac{5+2\sqrt{5}}{5}}
\end{align}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/163165",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Given number of trailing zeros in n!, find out the possible values of n. It's quite straightforward to find out number of trailing zeros in n!.
But what if the reverse question is asked?
n! has 13 trailing zeros, what are the possible values of n ?
How should we approach the above question ?
| Write $n$ in base $5$ as $n = a_0 + 5a_1 + 25a_2 + 125a_3 + \cdots$ where $0 \leq a_k \leq 4$. $$\left\lfloor \dfrac{n}5\right\rfloor = a_1 + 5a_2 + 25a_3 + \cdots$$
$$\left\lfloor \dfrac{n}{25}\right\rfloor = a_2 + 5a_3 + \cdots$$
$$\left\lfloor \dfrac{n}{125}\right\rfloor = a_3 + \cdots$$
Hence, $$\left\lfloor \dfrac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/163388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 1
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Solving a complex integral I need help solving an integral from John Conway book.
Lets $\alpha$ complex number different from 1 find integral $$\int\frac{dx}{1-2\alpha\cos{x}+{\alpha}^2}$$ from 0 to $2\pi$ in unit circle $$(z-\alpha)^{-1}(z-\frac{1}{\alpha})^{-1}$$
| Let $t=\tan(x/2) $ so that $\cos x = \frac{1-t^2}{1+t^2}$ and $dx=\frac{2 dt}{1+t^2}.$ The integral becomes $$ \int \frac{1}{1-2\alpha \frac{1-t^2}{1+t^2} + \alpha^2} \frac{2 dt}{1+t^2}=2 \int \frac{dx}{(\alpha-1)^2+ (\alpha+1)^2 t^2}= \frac{1}{\alpha^2-1} \tan^{-1} \left( \frac{(\alpha+1)t}{\alpha-1} \right)+ C= \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/164192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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A problem dealing with even perfect numbers. Question: Show that all even perfect numbers end in 6 or 8.
This is what I have. All even perfect numbers are of the form $n=2^{p-1}(2^p -1)$ where $p$ is prime and so is $(2^p -1)$.
What I did was set $2^{p-1}(2^p -1)\equiv x\pmod {10}$ and proceeded to show that $x=6$ or $... | Note that the powers of $2$ are congruent to $2$, $4$, $8$, or $6$, according to whether the exponent is congruent to $1$, $2$, $3$, or $0$ modulo $4$.
Assume $p\equiv 1\pmod{4}$. Then $2^{p-1}\equiv 6\pmod{10}$, and $2^p-1\equiv 1\pmod{10}$, so the product is congruent to $6$ modulo $10$.
If $p\equiv 3\pmod{4}$, then ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/168504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
How to calculate $\int_{-a}^{a} \sqrt{a^2-x^2}\ln(\sqrt{a^2-x^2})\mathrm{dx}$ Well,this is a homework problem.
I need to calculate the differential entropy of random variable
$X\sim f(x)=\sqrt{a^2-x^2},\quad -a<x<a$ and $0$ otherwise. Just how to calculate
$$
\int_{-a}^a \sqrt{a^2-x^2}\ln(\sqrt{a^2-x^2})\,\mathrm{d}x
... | This integral can be performed via differentiation under the integral sign. First note that for $|x|\leq1$ we have $\ln \sqrt{1-x^2} = \frac12\ln (1-x^2)$. Moreover, simple application of the chain rule yields
$$ \frac{d}{d\alpha} (1-x^2)^\alpha = (1-x^2)^\alpha \ln(1-x^2) .$$
The remaining integral is a special case o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/168686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
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Finding two numbers given their sum and their product
Which two numbers when added together yield $16$, and when multiplied together yield $55$.
I know the $x$ and $y$ are $5$ and $11$ but I wanted to see if I could algebraically solve it, and found I couldn't.
In $x+y=16$, I know $x=16/y$ but when I plug it back in... | We are trying to solve the system of equations $x+y=16$, $xy=55$. Here are a couple of systematic approaches that work in general.
Approach $1$: We will use the identity $(x+y)^2-4xy=(x-y)^2$. In our case, we have $(x+y)^2=256$, $4xy=220$, so $(x-y)^2=36$, giving $x-y=\pm 6$.
Using $x+y=16$, $x-y=6$, we get by adding ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 9,
"answer_id": 6
} |
Find all real solutions to $8x^3+27=0$
Find all real solutions to $8x^3+27=0$
$(a-b)^3=a^3-b^3=(a-b)(a^2+ab+b^2)$
$$(2x)^3-(-3)^3$$ $$(2x-(-3))\cdot ((2x)^2+(2x(-3))+(-3)^2)$$ $$(2x+3)(4x^2-6x+9)$$
Now, to find solutions you must set each part $=0$. The first set of parenthesis is easy $$(2x+3)=0 ; x=-\left(\frac{3}{... | You are working too hard. Note that
$$8x^3+27=0\iff x^3=\frac{-27}{8}\iff x=-\sqrt[3]{27/8}\iff x=-\frac{3}{2}$$
and so the only real solution is $x=-3/2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/171682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
Solve for $x$; $\tan x+\sec x=2\cos x;-\infty\lt x\lt\infty$
Solve for $x$; $\tan x+\sec x=2\cos x;-\infty\lt x\lt\infty$
$$\tan x+\sec x=2\cos x$$
$$\left(\dfrac{\sin x}{\cos x}\right)+\left(\dfrac{1}{\cos x}\right)=2\cos x$$
$$\left(\dfrac{\sin x+1}{\cos x}\right)=2\cos x$$
$$\sin x+1=2\cos^2x$$
$$2\cos^2x-\sin x... | You correctly applied the definitions of $\tan$ and $\sec$ to go from the equation in the problem statement to
$$\left(\dfrac{\sin x}{\cos x}\right)+\left(\dfrac{1}{\cos x}\right)=2\cos x.$$
However, you combined the two fractions incorrectly; in general
$$\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}$$
For example, you know t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
} |
Proving:$\tan(20^{\circ})\cdot \tan(30^{\circ}) \cdot \tan(40^{\circ})=\tan(10^{\circ})$ how to prove that : $\tan20^{\circ}.\tan30^{\circ}.\tan40^{\circ}=\tan10^{\circ}$?
I know how to prove
$ \frac{\tan 20^{0}\cdot\tan 30^{0}}{\tan 10^{0}}=\tan 50^{0}, $
in this way:
$ \tan{20^0} = \sqrt{3}.\tan{50^0}.\tan{10^0}$
... | In a word, yes. You already know that (in degrees) $\tan 20\cdot\tan30=\tan10\cdot\tan50$ so
$$\tan20\cdot\tan30\cdot\tan40 = \tan10\cdot\tan50\cdot\tan40$$
and your observation that
$$\frac{1}{\tan40}=\tan50$$
is all you need.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/172182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 0
} |
show that $x^2+y^2=z^5+z$ Has infinitely many relatively prime integral solutions How to show that this equation:
$$x^2+y^2=z^5+z$$
Has infinitely many relatively prime integral solutions
| The number $z^4+1$ is a sum of two relatively prime squares. Let $z$ be the sum of two relatively prime squares. Then the product $(z^4+1)z$ is a sum of two squares, by the Brahmagupta identity
$$(s^2+t^2)(u^2+v^2)=(su\pm tv)^2+(sv\mp tu)^2.$$
Now we take care of the relatively prime part. Suppose that $m$ has a repre... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/172399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Prove that $\left \{ \frac{n}{n+1}\sin\frac{n\pi}{2} \right \}$ is divergent. I want to prove whether the sequence $\{a_n\} = \left \{ \dfrac{n}{n+1}\sin\dfrac{n\pi}{2} \right \}$ (defined for all positive integers $n$) is divergent or convergent.
I suspect that it diverges, because the $\sin\frac{n\pi}{2}$ factor osci... | Here is another approach:
show that there exists infinitely many terms within $\delta$ neighbourhood of 0,1 and -1.
in other words 0,1 and -1 are all limits so there is no single limit for this sequence.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/173445",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Evaluating $\int_{|z|=1} \frac{\sin(z^2)}{ \left( \sin(z) \right)^2} dz.$ Consider $z \in \mathbb{C}$ and
$$\int_{|z|=1} \frac{\sin(z^2)}{ \left( \sin(z) \right)^2} dz.$$
How would we integrate this?
| $$\text{If}\,\,f(z)=\frac{\sin z^2}{\sin^2z}\,\,,\,\text{then}\,\,Res_{z=0}(f)=\lim_{z\to0}\frac{d}{dz}\left(z^2\frac{\sin z^2}{\sin^2z}\right)=0$$
So the singularity at $\,z=0\,$ is in fact a removable one and thus the integral equals zero.
We can also use power series in a tiny neighbourhood of zero:
$$\frac{\sin z^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/174379",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Minimum value of given expression What is the minimum value of the $$ \frac {x^2 + x + 1 } {x^2 - x + 1 } \ ?$$
I have solved by equating it to m and then discriminant greater than or equal to zero and got the answer, but can algebraic manipulation is possible
| For $x\geq0$ we have $$\frac{x^2+x+1}{x^2-x+1}=1+\frac{2x}{x^2-x+1}\geq1.$$
For $x<0$ by AM-GM we obtain:
$$\frac{x^2+x+1}{x^2-x+1}=1+\frac{2x}{x^2-x+1}=1+\frac{2}{x+\frac{1}{x}-1}=$$
$$=1-\frac{2}{-x+\frac{1}{-x}+1}\geq1-\frac{2}{2\sqrt{-x\cdot\frac{1}{-x}}+1}=\frac{1}{3}.$$
The equality occurs for $-x=\frac{1}{-x}$ o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/174905",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Gradient And Hessian Of General 2-Norm Given $f(\mathbf{x}) = \|\mathbf{Ax}\|_2 = (\mathbf{x}^\mathrm{T} \mathbf{A}^\mathrm{T} \mathbf{Ax} )^{1/2}$,
$\nabla f(\mathbf{x}) = \frac {\mathbf{A}^\mathrm{T} \mathbf{Ax}} {\|\mathbf{Ax}\|_2} = \frac {\mathbf{A}^\mathrm{T} \mathbf{Ax}} {(\mathbf{x}^\mathrm{T} \mathbf{A}^\mathr... | It is easier to work with $\phi(x) = \frac{1}{2} f^2(x)$. Just expand $\phi$ around $x$.
$\phi(x+\delta) = \frac{1}{2} (x + \delta)^T A^T A (x + \delta) = \phi(x) + x^TA^TA \delta + \frac{1}{2} \delta^T A^T A \delta$. It follows from this that the gradient $\nabla \phi(x) = A^T A x$, and the Hessian is $H = A^TA$.
To f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/175263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
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Proving Quadratic Formula purplemath.com explains the quadratic formula. I don't understand the third row in the "Derive the Quadratic Formula by solving $ax^2 + bx + c = 0$." section. How does $\dfrac{b}{2a}$ become $\dfrac{b^2}{4a^2}$?
| I see no visual proof, so I will add one.
The rectangle below is broken up into to parts: a square and another rectangle. Note that the area of the original rectangle is given by the sum of the two smaller areas: $$x^2 + bx$$
The next step is to divide the smaller vertical rectangle by two, rotate the strip and add it... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/176439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 5
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What's wrong with this conversion? I need to calculate the following limes:
$$
\lim_{n\rightarrow\infty} \sqrt{\frac{1}{n^2}+x^2}
$$
My first intuition was that the answer is $x$, but after a bit of fiddling with the root I got thoroughly confused. I know that below conversion goes wrong somwhere, but where?
$$
\lim_{n... | $$
\lim_{n\rightarrow\infty} \sqrt{\frac{1}{n^2}+x^2}
= \lim_{n\rightarrow\infty} \sqrt{\frac{1+x^2 \cdot n^2}{n^2}}
= \lim_{n\rightarrow\infty} \frac{\sqrt{1+x^2 \cdot n^2}}{n}
\neq \lim_{n\rightarrow\infty} \frac{\sqrt{\frac{1}{n^2}+x^2}}{n}
$$
$$
\lim_{n\rightarrow\infty} \sqrt{\frac{1}{n^2}+x^2}
= \lim_{n\rightarro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/177182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove $\cos^2 x \,\sin^3 x=\frac{1}{16}(2 \sin x + \sin 3x - \sin 5x)$ How would I prove the following?
$$\cos^2 x \,\sin^3 x=\frac{1}{16}(2 \sin x + \sin 3x - \sin 5x)$$
I do not know how to do do the problem I do know $\sin(3x)$ can be $\sin(2x+x)$ and such yet I am not sure how to commence.
| We know $e^{iy}=\cos y+i\sin y$
So, $\cos 5x+i\sin 5x=e^{i5x}=(e^{ix})^5=(\cos x+i \sin x)^5$
Using binomial theorem & equating imaginary parts,
$sin 5x = sin^5x − 10\cdot sin^3x cos^2x + 5\cdot sin x cos^4x$
$=sin^5x − 10\cdot sin^3x(1 - sin^2x) + 5\cdot sin x (1 - sin^2x)^2$
So, $sin 5x= 16\cdot\sin^5x − 20\cdot\sin^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/179294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
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How many positive values of $a$ are possible in $2^{3}\le a\lfloor a\rfloor \le 4^{2} + 1$ How many positive values of $a$ are possible in the following case?
$$2^{3}\le a\lfloor a\rfloor \le 4^{2} + 1$$
where $a\lfloor a\rfloor$ such that $a[a]$ is an integer.
| It is clear that $\lfloor a\rfloor$ must be $\ge 3$ and $\le 4$.
In the case $\lfloor a\rfloor=3$, the product can be $9$, $10$, or $11$. For $8$ is too small, since $a\ge \lfloor a\rfloor$. And $12$ is too big, since then $a=4$, giving the wrong value for the floor function. So $a$ can have values $3$, $10/3$, and $1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/181092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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$\frac{1}{x}+\frac{4}{y} = \frac{1}{12}$, where $y$ is and odd integer less than $61$. Find the positive integer solutions (x,y). $\frac{1}{x}+\frac{4}{y} = \frac{1}{12}$, where $y$ is and odd integer less than $61$.
Find the positive integer solutions (x,y).
| $y=\frac{48x}{x-12}$
Now $x-12$ must be multiple of 16, else y will be even.
Let x-12=16k=>$y=\frac{48(16k+12)}{16k}=>\frac{3(16k+12)}{k}=48+\frac{36}{k}$
So, k must divide 36 and must be of the form 4r, where r is an odd integer.
$1≤y≤61=>1≤48+\frac{36}{k}≤61$
$\frac{36}{k}≤13=>k≥3$.
But $16k=x-12=>x=12+16k>0=>k>-1$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/182882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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Spivak Calculus 3rd ed. $|a + b| \leq |a| + |b|$ I'm working through the first chapter of Michael Spivak's Calculus 3rd ed.
Towards the end of the chapter he proves $ |a + b| ≤ |a| + |b| $ using the observation that $|a|= \sqrt{ a^2 }$ when $a$ is $ ≥ 0 $ .
$ |a + b| ≤ |a| + |b| $
$$ (|a + b|)^2 = (a + b)^2 $$ $$= a... | Let $a=r_1(\cos A+i\sin A)$ and $b=r_2(\cos B+i\sin B)$
So, $|a|=r_1$ and $|b|=r_2$
Now, $|a+b|=\sqrt{(r_1\cos A + r_2\cos B)^2+(r_1\sin A + r_2\sin B)^2}$
$=\sqrt{r_1^2+r_2^2+2r_1r_2\cos(A-B)}$
$≤\sqrt{r_1^2+r_2^2+2r_1r_2}\ $ as $\cos(A-B)≤1$ for real A,B
$=r_1+r_2=|a|+|b|$
Also observe, $|a+b|=\sqrt{r_1^2+r_2^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/183520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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How to prove Lagrange trigonometric identity I would to prove that
$$1+\cos \theta+\cos 2\theta+\ldots+\cos n\theta =\displaystyle\frac{1}{2}+
\frac{\sin\left[(2n+1)\frac{\theta}{2}\right]}{2\sin\left(\frac{\theta}{2}\right)}$$
given that
$$1+z+z^2+z^3+\ldots+z^n=\displaystyle\frac {1-z^{n+1}}{1-z}$$
where $z\neq 1$... | Here's a variation of @lab's variation of @DonAntonio's solution:
$$\begin{align}
\frac{e^{(n+1)i\theta}-1}{e^{i\theta-1}-1}
&= \frac{e^{(n+1)i\theta}-1}{e^{i\theta/2} \left(e^{i\theta/2}-e^{-i\theta/2}\right)} \\
&= \frac{e^{(n+1/2)i\theta}-e^{-i\theta/2}}{2i\sin\frac{\theta}{2}} \\
&= \frac{\mathrm{cis}\frac{(2n+1)i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/183859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 2
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Prove the inequality $a^2bc+b^2cd+c^2da+d^2ab \leq 4$ with $a+b+c+d=4$ Let $a,b,c$ and $d$ be positive real numbers such that $a+b+c+d=4.$
Prove the inequality
$$a^2bc+b^2cd+c^2da+d^2ab \leq 4 .$$
Thanks :)
| Let $\{a,b,c,d\}=\{x,y,z,t\}$, where $x\geq y\geq z\geq t$.
Hence, since $(x,y,z,t)$ and $(xyz,xyt,xzt,yzt)$ are the same ordered,
by Rearrangement and AM-GM we obtain:
$$a^2bc+b^2cd+c^2da+d^2ab=a\cdot abc+b\cdot bcd+c\cdot cda+d\cdot dab\leq$$
$$\leq x\cdot xyz+y\cdot xyt+z\cdot xzt+t\cdot yzt=xy(xz+yt)+zt(xz+yt)=$$
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/184266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Finding all integers such that $a^2+4b^2 , 4a^2+b^2$ are both perfect squares Are there any integers $a,b$, such that:
$$a^2+4b^2 , 4a^2+b^2$$
are both perfect squares?
| Integers $m$ such that,
$$ma^2+b^2=d^2\\a^2+mb^2 = c^2\tag1$$
have solutions unfortunately do not have an official name. However, the $n$ of a similar system,
$$a^2+b^2 = c^2\\a^2+nb^2=d^2\tag2$$
are called concordant numbers and it is well-established it has positive solutions $a,b$ for the sequence,
$$n=1, 7, 10, 11,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/184659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
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Prove inequality $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{9\sqrt[3]{abc}}{a+b+c}\geq 6$ Let $a,b,c>0$.
What is the proof that:
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{9\sqrt[3]{abc}}{a+b+c}\geq 6$$
| Take
$$ \frac{a}{b}+\frac{a}{b}+\frac{b}{c}\geq 3\frac{a}{ (abc)^\frac{1}{3}}$$
$$ \frac{b}{c}+\frac{b}{c}+\frac{c}{a}\geq 3\frac{b}{ (abc)^\frac{1}{3}}$$
$$ \frac{c}{a}+\frac{c}{a}+\frac{a}{b}\geq 3\frac{c}{ (abc)^\frac{1}{3}}$$
from AM-GM
and then add them and you get
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \geq \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/185604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 1,
"answer_id": 0
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Finding the sum of this alternating series with factorial denominator. What is the sum of this series?
$$ 1 - \frac{2}{1!} + \frac{3}{2!} - \frac{4}{3!} + \frac{5}{4!} - \frac{6}{5!} + \dots $$
| Alternatively, write it as:
$$1-\frac{1}{1!} +\frac{1}{2!} - \frac{1}{3!}... +\\
\left(-\frac{1}{1!}+ \frac{2}{2!} - \frac{3}{3!}...\right)$$
The first line is $e^{-1}$ and the second line, after cancelling terms, you see is $-e^{-1}$
More generally, if $$(z)_i = z(z-1)...(z-(i-1))$$ is the falling factorial, and $p(z)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/185915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Inequality:$ (a^{2}+c^{2})(a^{2}+d^{2})(b^{2}+c^{2})(b^{2}+d^{2})\leq 25$ For $ a,b,c,d\geq 0 $ with $ a+b = c+d = 2 $, how to prove that $$ (a^{2}+c^{2})(a^{2}+d^{2})(b^{2}+c^{2})(b^{2}+d^{2})\leq 25$$
| We will make repeated use of the identity $(p^2+q^2)(r^2+s^2)=(pr-qs)^2+(ps+qr)^2$.
Applying it to the first two terms yields
$$
(a^2+c^2)(a^2+d^2)=(a^2-cd)^2+(a(c+d))^2=(a^2-cd)^2+(2a)^2 \, .
$$
Similarly, the third and fourth terms multiply to $(b^2-cd)^2+(2b)^2$.
Applying the identity again to these two quantities, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/186437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 1,
"answer_id": 0
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Second order non-linear differential equation $ y_n'' -nx\frac{1}{\sqrt {y_n}}=0$
Is there any known method to solve such second order non-linear differential equation?
What I tried to solve:
$ 2y_n'y_n''= 2nx\frac{y_n'}{\sqrt {y_n}}$
$ y_n'^2= 4nx\sqrt {y_n}-4n\int\sqrt {y_n} dx$
After that I could not see any way how... | In fact it belongs to an Emden-Fowler equation.
First, according to http://eqworld.ipmnet.ru/en/solutions/ode/ode0302.pdf or http://www.ae.illinois.edu/lndvl/Publications/2002_IJND.pdf#page=6 , all Emden-Fowler equations can be transformed into Abel equation of the second kind.
Let $\begin{cases}u=\dfrac{x^3}{y_n^\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/188277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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how can one find the value of the expression, $(1^2+2^2+3^2+\cdots+n^2)$
Possible Duplicate:
Proof that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$?
Summation of natural number set with power of $m$
How to get to the formula for the sum of squares of first n numbers?
how can one find the value of the expressio... | In general, $$\sum_{i=1}^{n}i^{2} = \frac{n(n+1)(2n+1)}{6}.$$
A collection of proofs of this fact can be found here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/188602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Proving inequality $\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}} \leq \sqrt{3 \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)}$ In the pdf which you can download here I found the following inequality which I can't solve it.
Exercise 2.1.11 Let $a,b,c \gt 0$. Prove that
$$\sqrt{\frac{2a}{b+c}}... | By C-S
$$\left(\sum_{cyc}\sqrt{\frac{a}{b+c}}\right)^2\leq\sum_{cyc}\frac{a}{a+c}\sum_{cyc}\frac{a+c}{b+c}.$$
Thus, it remains to prove that
$$\frac{3}{2}\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\geq\sum_{cyc}\frac{a}{a+c}\sum_{cyc}\frac{a+c}{b+c}$$ or
$$\sum_{cyc}(3a^6c^3+3a^5b^4+6a^5c^4+a^6b^2c+2a^6c^2b+4a^5b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/189140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Sum of product of Fibonacci numbers I want to calculate the sum of product of Fibonacci number for a given $n$. That is, for given $n$, say
$$F_1 F_n + F_2 F_{n-1} + F_3 F_{n-2} + F_4 F_{n-3} + F_5 F_{n-4} + \cdots.$$
what should be my approach.
| Let's use
$$
F_n = \frac{1}{\sqrt{5}} \left( \phi^n - (-1)^n \phi^{-n} \right)
$$
where $\phi$ is a Golden ratio constant $\phi = \frac{\sqrt{5}+1}{2}$. Now the sum reduces to the combination of geometric sums:
$$
\begin{eqnarray}
\sum_{k=1}^{n-1} F_k F_{n-k} &=& \frac{1}{5}\sum_{k=1}^{n-1} \left(\phi^k - (-1)^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/190443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Solve $\sqrt{x-4} + 10 = \sqrt{x+4}$ Solve: $$\sqrt{x-4} + 10 = \sqrt{x+4}$$
Little help here? >.<
| We will assume that $x$ ranges over the reals $\ge 4$, to make sure that the square roots are real. Note that
$$\sqrt{x+4}-\sqrt{x-4}=\frac{(\sqrt{x+4}-\sqrt{x-4})(\sqrt{x+4}+\sqrt{x-4})}{\sqrt{x+4}+\sqrt{x-4}} =\frac{8}{\sqrt{x+4}+\sqrt{x-4}} .$$
For $x\ge 4$, $\sqrt{x+4}+\sqrt{x-4}\ge 2\sqrt{2}$. It follows that $\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/192125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 1
} |
How to solve $x^3=-1$? How to solve $x^3=-1$? I got following:
$x^3=-1$
$x=(-1)^{\frac{1}{3}}$
$x=\frac{(-1)^{\frac{1}{2}}}{(-1)^{\frac{1}{6}}}=\frac{i}{(-1)^{\frac{1}{6}}}$...
| Set $\displaystyle x=re^{i \theta}$. So $\displaystyle r^3e^{i3\theta}= x^3= -1= e^{i \pi}$, hence $r^3=1$ and $3 \theta= \pi [2\pi]$. Finally, $r=1$ and $\displaystyle \theta = \frac{\pi}{3} \left[\frac{2\pi}{3} \right]$ so you get three solutions: $$x_1= e^{i \pi/3}= \frac{1}{2}+ i \frac{\sqrt{3}}{2}, \ x_2=e^{-i 2 \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/192742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
"answer_count": 8,
"answer_id": 0
} |
Subsets and Splits
Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.