Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Limit proof, finding upper bounds I need to prove that $\frac{1}{x-3} \to 1$ as $x \to 4$
So I did $\left | \frac{1}{x-3} - 1 \right | = \left | \frac{4 - x}{x-3} \right | = \left | \frac{x-4}{x-3} \right | = \frac{|x-4|}{|x-3|} < K|x-4|$. So I need to pick $| x - 4| < \delta = \frac{\epsilon}{K}$
Now the problem is... | Let $\epsilon>0$. Suppose $|x-4|<1/2$. Then $-1/2<x-4<1/2$ which implies that $1/2<x-3<3/2$. Hence, $|x-3|\ge x-3>1/2$, that is, if $x\ne 3$ then $\frac{1}{|x-3|}<2$. Define $\delta=\min\left\{\frac{1}{2},\frac{\epsilon}{2}\right\}$. Let $0<|x-4|<\delta$. Then certainly $x\ne 3$. Hence,
$\left| \frac{1}{x-3}-1\right|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/252830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Using formula of sum of two squares ... create Pythagorean triplets from pairs of primitive Pythagorean triplets? Can you use a formula for writing products of two sums of squares as a sum of squares to create Pythagorean triplets from pairs of primitive Pythagorean triplets?
First, the formula for writing products of ... | Let the following be your Pythagorean triplets.
$$a_1^2 +b_1^2 = c_1^2$$
$$a_2^2 +b_2^2 = c_2^2$$
We have (from the formula):
$$(a_1^2 +b_1^2)\cdot(a_2^2 +b_2^2)=(a_1\cdot b_2-a_2\cdot b_1)^2+(a_1\cdot a_2-b_1\cdot b_2)^2$$
Obviously the RHS is the sum of two squares. To prove this is a Pythagorean triplet, we need to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/255073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Sum of special series: $1/(1\cdot2) + 1/(2\cdot3 )+ 1/(3\cdot4)+\cdots$ The question is:
Find the sum of the series $$ 1/(1\cdot 2) + 1/(2\cdot3)+ 1/(3\cdot4)+\cdots$$
I tried to solve the answer and got the $n$-th term as $1/n(n+1)$. Then I tried to calculate $\sum 1/(n^2+n)$. Can you help me?
|
Find the sum of the series $ \frac{1}{1\cdot 2} + \frac{1}{2\cdot3}+ \frac{1}{3\cdot4}+\cdots +\frac{1}{n(n+1)}$
You have a series called Telescoping series as M. Strochyk pointed out in the comment. Its $i^{th}$ term is $\frac{1}{i(i+1)}$ and $n^{th}$ term is $\frac{1}{n(n+1)}$ as shown in the series. Thus,
$$
\sum_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/255306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
A question on complex numbers We are given
If $\cos(a+ib)$=$r (\cos\theta +i\sin\theta)$
then prove that $e^{2b} = \sin(a-\theta)/\sin(a+\theta)$
I just tried and got $b = 0$ such that $\cos(a) = ra$. Will there be other solutions?
| $\displaystyle \cos (a+ib) = \cos a \cos ib - \sin a \sin ib = \cos a \cosh b - i \sin a \sinh b = r (\cos \theta + i \sin \theta) \\
\displaystyle r \cos \theta = \cos a \cosh b = \cos a \frac {e^{2b} + 1}{2e^b}\\
\displaystyle r \sin \theta = -\sin a \sinh b = -\sin a \frac {e^{2b} - 1}{2e^b} \\
\displaystyle \tan \t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/256658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
If $(a,b)=1$ then prove $(a+b, ab)=1$.
Let $a$ and $b$ be two integers such that $\left(a,b\right) = 1$. Prove that $\left(a+b, ab\right) = 1$.
$(a,b)=1$ means $a$ and $b$ have no prime factors in common
$ab$ is simply the product of factors of $a$ and factors of $b$.
Let's say $k\mid a+b$ where $k$ is some factor of... | HINT: Suppose that $p$ is a prime that divides $ab$ and $a^2+b^2$. Then $p$ divides both $a^2+2ab+b^2=(a+b)^2$ and $a^2-2ab+b^2=(a-b)^2$. This in turn implies that $p$ divides both $a+b$ and $a-b$. (Why?) Use this to show that $p$ divides both $a$ and $b$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/257434",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 16,
"answer_id": 3
} |
Proving a number defined by a sequence is a square number I found this problem in a math magazine:
Given the sequence $(x_n)_{n \in \mathbb{N}}$ defined by:
$$
x_0 = 0\\
x_1 = 1\\
x_{n+2}+x_{n+1}+2x_{n}=0
$$
Prove that $s_n = 2^{n+1}-7x_{n-1}^2, n > 0$ is a square number.
I tried searching a rule between the numb... | This may be a little brute force, but it's a solution:
If you're familiar with solving linear recurrences, you'll know that you can write a formula for the $x_n$ of the form $$x_n = A\lambda_+^n + B\lambda_-^n,$$ where $A$ and $B$ are constants and the $\lambda_\pm$ are the roots of the polynomial $x^2 + x + 2$, that i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/259305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
} |
show that if $a,b,c \in \mathbb{R}^+$ show that if $a,b,c \in \mathbb{R}^+$ different from zero, then:
$$(a^2+b^2+c^2)\cdot\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)\leq(a+b+c)\cdot\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$$
I had no success in my attempts
| $a=2012,b=c=1$.
Any other large number should work instead of $2012$.
If the inequality is reversed, just multiply and prove the following simple Lemma:
Lemma $f(x) =x+ \frac{1}{x}$ is increasing on $[1, \infty)$. In particular, for all $x\geq 1$ we have
$$f(x^2) \geq f(x) \,.$$
P.S. Maybe even simpler
$$\frac{a}{b}+\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/261669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How can I solve quadratic equations using modular arithmetic How can I solve quadratic equations using modular arithmetic? E.g.
$$2x^2 + 8x + 2 = 0 \pmod{23}$$
N.b. I have changed the figures from those in my homework question because I don't want a solution I want to understand the process. Consequently the example I ... | We have $2x^2+8x+2\equiv 0\pmod{23}$ if and only if $x^2+4x+1\equiv 0\pmod{23}$.
Now complete the square. We have $x^2+4x+1=(x+2)^2-3$. So we want to solve the congruence $(x+2)^2\equiv 3\pmod{23}$.
Let $y=x+2$. We want to solve the congruence $y^2\equiv 3\pmod{23}$.
There is general theory that, for large $p$, helps ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/261896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Fourier transform of $f(x)=\frac{1}{x^2+6x+13}$ How to find the Fourier transform of the following function:
$$f(x)=\frac{1}{x^2+6x+13}$$
| This begs for completing the square:
$$
x^2 + 6x + 13 = \Big( x^2 + 6x + 9 \Big) + 4 = (x+3)^2 + 4.
$$
Then
$$
\frac{1}{(x+3)^2 + 4} = \frac 1 4 \cdot \frac{1}{\left(\dfrac{(x+3)^2}{4}\right) + 1} =\frac 1 4 \cdot \frac{1}{\left(\frac{x+3}{2}\right)^2+1} = \frac 1 4 \cdot \frac{1}{w^2 + 1}
$$
The Laplace transform inv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/262321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 4
} |
How to solve system of equations? I want to sove the system of equations $$\begin{cases}
x^3 y-y^4=7,\\
x^2 y+2 xy^2+y^3=9.
\end{cases}
$$
I tried divide these two equations we obtain
$$\dfrac{x^3 - y^3}{(x+y)^2 } = \dfrac{7}{9}$$
From here, I don't know how to solve.
| We'll prove that $(2,1)$ is the only real solution. First note that if $y = 0$, the second equation is unsolvable, so $y\neq 0$.
Solving the second equation for $x$ via the quadratic formula gives \begin{align*} x &= \frac{-2y^2 \pm \sqrt{4y^4-4y(y^3-9)}}{2y}\\ &= \frac{-y^2\pm 3\sqrt{y}}{y} \\ &= -y \pm y^{-\frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/262449",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Four digit reversal numbers How to prove without an exhaustive checking that there are only 2 (nontrivial) four digit reversal numbers?
| Let $A = \{2,3,4,5,6,7,8,9\}$.
Let the $4$ digit reversal number be $abcd$ i.e. $1000a + 100b + 10 c + d$, where $a \in \{1\} \cup A$, $d \in \{1,2,3,\ldots,a-1\}$ and $b,c \in \{0,1\} \cup A$. Its reversed number is $$dcba = 1000d + 100c + 10b + a$$ We want $abcd = k \times (dcba)$ where $k \in A$. This gives us
$$(10... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/263429",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Let $\{a_{n}\}$ be any sequence of reals such that $\lim \limits_{n\rightarrow \infty} na_{n} =0$... Let $\{a_{n}\}$ be any sequence of reals such that $\lim \limits_{n\rightarrow \infty } na_{n} =0$. Prove that $$\lim_{n \rightarrow \infty } \left( 1 + \frac{1}{n} + a_{n}\right)^{n} =e$$
I think I have shown that $\... | First, let's factor out the part we know, then simplify
$$ \begin{align}
\lim_{n \rightarrow \infty } \left( 1 + \frac{1}{n} + a_{n}\right)^{n}
&= \lim_{n \rightarrow \infty } \left( 1 + \frac{1}{n} \right)^{n}
\left( \frac{1 + \frac{1}{n} + a_{n}}{1 + \frac{1}{n}} \right)^{n}
\\ &= \lim_{n \rightarrow \infty } \left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/264452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
product of two distinct squares Is there any shorter and efficient way to find
if a number can be formed by the product of two distinct square numbers
for example
*
*36=4*9
*144=16*9
help me with an algorithm or the logic
| You are looking for a number formed by product of two squares, so you have it written in this form: $n = a^2 b^2$, apply product of powers property, so you have: $n = a^2 b^2 = (ab)^2$, so at the end you have to check if this number is a square. If you want to control if the number is made of two factors you have to fi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/265336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Calculating that 13 is a Wilson Prime I'm confused by the idea of a Wilson Prime. The theorem states that $$p^2=(p-1)!+1$$
This makes sense for $5$: $$5^2=(4\times3\times2)+1$$ so $5^2=25$
But it makes no sense to me for $13$: $$13^2=4790016001$$
Clearly I am as far from a mathematician as possible. If you can help in... | To prove that $13$ is a Wilson prime, you need to show that
$$12! \equiv -1 \pmod {13^2} \,.$$
To make the computations faster, observe that
$$12!= 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot (2 \cdot 5)\cdot 11 \cdot (4 \cdot 3)$$
We split the product in three:
$$2 \cdot 3 \cdot 4 \cdot 7 =(2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/266060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Prime sum identity Let $ \Lambda(k) $ denote the von Mangoldt function:
$$
\Lambda(k) \stackrel{\text{def}}{=}
\begin{cases}
0 & \text{if $ k $ is not a prime power}, \\
\ln(p) & \text{if $ k = p^{j} $}.
\end{cases}
$$
Also, let $ \lfloor x \rfloor $ be the floor function.
Can anyone prove any of the following ide... | A perhaps simpler proof, using the $p$-adic expansion of $n!$:
$$\log(n!)=\sum_{p \text{ prime}}\log(p)\left(\left \lfloor \frac{n}{p} \right \rfloor+\left \lfloor \frac{n}{p^2} \right \rfloor+...\right)=\sum_{k=1}^n \Lambda(k)\left \lfloor \frac{n}{k} \right \rfloor. $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/267439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
How to prove this inequality with the following condition? Let $x$, $y$, $z$ be three positive real numbers satisfying
\begin{equation}
x + y +z + 1 =4xyz.\tag{1}
\end{equation}
Prove that
\begin{equation}
xy + yz + zx \geqslant x + y + z.\tag{2}
\end{equation}
I don't know how to start?
| Let $x+y+z=3u$, $xy+xz+yz=3v^2$ and $xyz=w^3$.
Hence, the condition gives $3u+1=4w^3$, which does not depend on $v^2$.
We need to prove that $v^2\geq u$, for which it's enough to prove it for a minimal value of $v^2$.
In another hand $x$, $y$ and $z$ are positive roots of the following equation.
$$X^3-3uX^2+3v^2X-w^3=0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/272830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Find the standard form of the conic section $x^2-3x+4xy+y^2+21y-15=0$ Find the standard form of the conic section $x^2-3x+4xy+y^2+21y-15=0$.
I understand the approach in trying to solve these problems. But the $4xy$ is confusing me. I am not sure of where to start on this one.
UPDATE:
i have found a way of doing this w... | Consider the second degree terms: $x^2 + 4xy + y^2$. In general, if the second degree term is $Ax^2 + Bxy + Cy^2$, then the discriminant is $D = B^2 - 4AC$. If the conic section is non-degenerate, then
\begin{align}
D > 0 & \implies \text{Hyperbola}\\
D = 0 & \implies \text{Parabola}\\
D < 0 & \implies \text{Ellipse}
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/273308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Inspecting the function $f(x)=-x\sqrt{1-x^2}$ We are just wrapping up the first semester calculus with drawing graphs of functions. I sometimes feel like my reasoning is a bit shady when I am doing that, so I decided to ask you people from Math.SE.
I am supposed to draw a graph (and show my working) of the function $f... | Your answer and/or analysis of the function $$f(x)=-x\sqrt{1-x^2}$$ is accurate, thorough, well-justified, and consistent with its graph: $\quad \quad f(x)=-x\sqrt{1-x^2}$.
$\quad$Source: Wolfram Alpha.
Kudos for the effort you've shown and your accurate and detailed analysis! (Just don't forget to graph the func... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/275276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Causal Inverse Z-Transform of Fibonacci Say the Fibonacci sequence is defined by:
$y(n) = y(n-1) + y(n-2)$
initial conditions: $y(0)=0, y(1)=1$
I incorporate those initial conditions as:
$y(n) = y(n-1) + y(n-2) + \delta(n-1)$
I compute the z-transform as:
$Y(z) = z^{-1}Y(z) + z^{-2}Y(z) + z^{-1}$
solve for $Y(z)... | Using partial fraction techniques and denoting the zeroes of $z^2-z-1$ by $\alpha_1$ and $\alpha_2$, we have
$$ Y(z) = \frac{z}{z^2-z-1} \implies \frac{Y(z)}{z} = \left(-\frac{1}{5}+\frac{2}{5}\alpha_1\right) \frac{1}{z-\alpha_1}+ \left(-\frac{1}{5}+\frac{2}{5}\alpha_2\right) \frac{1}{z-\alpha_2} $$
$$ \implies Y(z) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/279868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
What is the Maximum Value? If $a, b, c, d, e$ and $f$ are non negative real numbers such that $a + b + c + d + e + f = 1$, then what is the maximum value of $ab + bc + cd + de + ef$?
| Here's the full solution in case the comments weren't enough.
Note that $(a+c+e)(b+d+f) = (a+c+e)(1-(a+c+e)) \le \frac{1}{4},$ with equality iff $a + c + e = \frac{1}{2}.$ Expanding the first expression above gives $(ab+bc+cd+de+ef)+(ad+af+be+cf) \le \frac{1}{4}.$ Since all of our variables are non-negative, $ad + af +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/280478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
show $\ln \frac{1-x}{1+x}$ is $L^2$ but not $L^1$ using its taylor expansion I am trying to show that
$$
\ln{\Big|\frac{1-x}{1+x}\Big|}
$$
belongs to $L^2({\Bbb{R}})$ but not to $L^1({\Bbb{R}})$ by using it's taylor expansion (this is the entire statement of the problem). More important to me than the solution to thi... | Let's begin with
$$\int_{-\infty}^{\infty} \left ( \log{\Big|\frac{1-x}{1+x}\Big|} \right )^2 dx $$
Note that the integrand is an even function. You can then split this integral up into two pieces:
$$ 2 \int_{0}^{1} \left ( \log{\Big|\frac{1-x}{1+x}\Big|} \right )^2 dx + 2 \int_{1}^{\infty} \left ( \log{\Big|\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/281279",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Prove $\frac{a}{bc}+ \frac{b}{ca}+ \frac{c}{ab} \ge 1$ Let $a,b,c$ be positive real numbers such that $\dfrac{1}{bc}+ \dfrac{1}{ca}+ \dfrac{1}{ab} \ge 1$. Prove that $\dfrac{a}{bc}+ \dfrac{b}{ca}+ \dfrac{c}{ab} \ge 1$.
| Remember, that for any nonzero $x$, one has
$$
x+\frac{1}{x} \geq 2 \tag{1}
$$
We apply this inequality repeatedly. Put
$$
T=\dfrac{a}{bc}+ \dfrac{b}{ca}+ \dfrac{c}{ab}
$$
We have
$$
T=\frac{a^2+b^2}{abc}+\frac{c}{ab} =\frac{\sqrt{a^2+b^2}}{ab} \bigg(\frac{\sqrt{a^2+b^2}}{c}+\frac{c}{\sqrt{a^2+b^2}
}\bigg) \geq \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/281966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Integral help here please? I have to solve this
$$\int \frac{dx}{(x+1)\sqrt{x^2+2x}}$$
First make the substitution $u=x+1$ and get
$$\int \frac{ds}{u\sqrt{u^2-1}}$$
Next I substitute $s=\sqrt{u^2-1}$ and I get $ds=u\,du/\sqrt{u^2-1}$.
I have integral of $1/s^2+1\dots$ I replace for $s$ and I get the integral as
$... | You wanted $\;s=\sqrt{u^2-1}$, so $\;s^2=u^2-1$, $\;2s\,ds=2u\,du$, and $\;s\,ds=u\,du$.
Substituting gives us:
$$
\begin{align}\int\frac{du}{u\sqrt{u^2-1}} &= \int\frac{u\,du}{u^2 \sqrt{u^2-1}} \\&= \int\frac{s\,ds}{(s^2+1)s}\\&= \int\frac{ds}{s^2+1} \\ \\&= \arctan s + C.
\end{align}
$$
Now don't forget to "back subs... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/282898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 0
} |
Fibonacci Proof
Prove that: $$F_1F_2+F_2F_3+F_3F_4+\cdots+F_{2n-1}F_{2n}=F_{2n}^2$$
I set it up so:
$$F^2(2k) + F(2k+1)F(2k+2) = F^2(2k+2)$$
I've tried:
$$F(2k)^2 + F(2k+1)*F(2k+2) = F(2k+2)*F(2k)+F(2k+2)*F(2k+1)$$
$$F(2k)^2 = F(2k+2)*F(2k)$$
$$F(2k+1)*F(2k) - F(2k)*F(2k-1) = F(2k)*F(2k)+F(2k)*F(2k+1)$$
$$-F(2k)*F(2k... | We can avoid induction using Binet's Fibonacci Number Formula, $$F_n=\frac{a^n-b^n}{a-b}$$ where $a,b$ are the roots of $x^2-x-1=0\iff a+b=1,ab=-1$ also, $a^2-a-1=0$
$$F_rF_{r+1}=\frac{(a^r-b^r)}{(a-b)}\frac{(a^{r+1}-b^{r+1})}{(a-b)}=\frac{a^{2r+1}+b^{2r+1}-(ab)^r(a+b)}{(a-b)^2}$$
$$=\frac{a^{2r+1}+b^{2r+1}-(-1)^r}{(a-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/285985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 0
} |
Working with exponent on series Hi have this sequence:
$$\sum\limits_{n=1}^\infty \frac{(-1)^n3^{n-2}}{4^n}$$
I understand that this is a Geometric series so this is what I've made to get the sum.
$$\sum\limits_{n=1}^\infty (-1)^n\frac{3^{n}\cdot 3^{-2}}{4^n}$$
$$\sum\limits_{n=1}^\infty (-1)^n\cdot 3^{-2}{(\frac{3}{4}... | You have
$$\begin{align}
\sum\limits_{n=1}^\infty \frac{(-1)^n3^{n-2}}{4^n}
&= \sum_{n=1}^{\infty} \frac{(-1)(-1)^{n-1}\frac{1}{3}3^{n-1}}{4\cdot 4^{n-1}} \\
&= \sum_{n=1}^{\infty} \frac{-1}{4\cdot 3}\frac{(-1)^{n-1}3^{n-1}}{4^{n-1}}\\
&= \sum_{n=1}^{\infty} \frac{-1}{12}\left(\frac{-3}{4}\right)^{n-1}
\end{align}$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/287405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Ratio of areas of two triangles Based on the figure below, what is the ratio of the area of triangle $CGI$ to the area of triangle $ABC$, in terms of $\theta$?
| Mathematica isn’t needed for the simplification, which is quite straightforward:
$$\begin{align*}
\frac{\left({\sin\theta \over \cos\theta} - \sin\theta\tan\theta\right)^2 \tan\theta}{\sin\theta\big(\cos\theta+\sin\theta\tan\theta\big)}&=\frac{\left(\tan\theta-\sin\theta\tan\theta\right)^2\tan\theta}{\frac{\sin\theta}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/287935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
How to calculate $\sum_{n = 2}^{+\infty} \frac{5 + 2^n}{3^n}$? I need to calculate:
$$\sum_{n = 2}^{+\infty} \frac{5 + 2^n}{3^n}$$
Actually, I'm not particularly interested in knowing the result. What I'm really interested in is:
*
*What kind of series is it?
*How should I start calculating series like that?
| Your sum is the sum of two geometric series:
$$\sum_{n=2}^\infty \frac{5 + 2^n}{3^n} = 5\sum_{n=2}^\infty \left(\frac{1}{3}\right)^n + \sum_{n=2}^\infty \left(\frac{2}{3}\right)^n$$
In general, $$\sum_{n=0}^\infty ar^n = \frac{a}{1-r}$$ which converges when the magnitude of $r < 1$.
In your sum of two geometric series,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/290042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Asymptotic expansion of $ I_n = \int_0^{\pi/4} \tan(x)^n \mathrm dx $ I'm trying to compute the asymptotic expansion of
$$ I_n = \int_0^{\pi/4} \tan(x)^n \mathrm dx $$
Here is what I've done:
Change of variable $$ t= \tan x $$
$$ I_n = \int_0^1 \frac{t^n \mathrm dt}{1+t^2} = \int_0^1 \frac{(1-t)^n \mathrm dt}{t^2-2t+2}... | I figured I'd throw my hat into the ring as well.
We can appeal to Watson's lemma to find a full asymptotic expansion of the integral.
After your substitution $t=\tan x$, make another substitution $t = e^{-s}$. This gives
$$
\int_0^1 t^n \frac{dt}{1+t^2} = \int_0^\infty e^{-ns} \frac{ds}{e^s+e^{-s}}.
$$
By Watson's le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/290772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 0
} |
Floor function inequality I am racking my brain trying to get this problem solved and I can't seem to break it...
Let $m, n$ be positive integers, with $m > 1$. Prove
$$\left\lfloor\frac{n}m\right\rfloor+\left\lfloor\frac{n+1}m\right\rfloor\le\left\lfloor\frac{2n}m\right\rfloor$$
I started trying to use the inequaliti... | If you want to use you that inequality of yours let $x=\dfrac{n}{m}$ and $y=\dfrac{1}{m}:$
$$\bigg\lfloor \frac{n}{m}\bigg\rfloor+\bigg\lfloor \frac{1}{m}\bigg\rfloor+\bigg\lfloor \frac{n+1}{m}\bigg\rfloor\leq\bigg\lfloor \frac{2n}{m}\bigg\rfloor+\bigg\lfloor \frac{2}{m}\bigg\rfloor$$
Since $m>1\Rightarrow \bigg\lfloor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/295105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Another Fibonacci identity: $F_{2n-1} = F_{n}^2 + F_{n-1}^2$ Here's a problem that is leading me in circles.
Consider the Fibonacci number $F_n$ defined by $F_0 = 0$, $F_1 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for all $n \geq 2$.
Prove that $F_{2n-1} = F_{n}^2 + F_{n-1}^2$.
Tried an induction proof that lead me nowher... | The identity may be derived from the interesting fact that
$$\left ( \begin{array} \\ 1 & 1\\1 & 0 \\ \end{array} \right ) ^k = \left ( \begin{array} \\ F_{k+1} & F_k\\F_k & F_{k-1} \\ \end{array} \right ).$$
From this, we may observe that
$$\begin{align} \left ( \begin{array} \\ 1 & 1\\1 & 0 \\ \end{array} \right ) ^m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/295173",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Baricenter of $4$ intersection points of parabola with circle lies on axis of parabola Show that the baricenter of the $4$ intersection points of a parabola with a circle is on the axis of the parabola.
Let $p$ be a parabola, $c$ a circle and $p\cap c=\{P_1,P_2,P_3,P_4\} \Rightarrow B= \frac{P_1+P_2+P_3+P_4}{4} \in$ ax... | If we expand a fourth-degree polynomial function $x\mapsto(x-p)(x-q)(x-r)(x-s)$, we get
$$
x^4 - (p+q+r+s)x^3 + \cdots.
$$
The sum of the roots is minus the coefficient of $x^3$.
The $x$-coordinates of the intersection of the parabola $y=x^2$ with the circle $(x-h)^2+(y-k)^2 = r^2$ are the roots of
$$
(x-h)^2 + (x^2-k)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/297483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Define $x_{2n}=\frac{x_{2n-1}+2x_{2x-2}}{3}\,\,\,\,\,,x_{2n+1}=\frac{2x_{2n}+x_{2n-1}}{3}$ for $n \in \mathbb{N^+}$. Let $x_0=a$ and $x_1=b$ with $b>a$. Define $$x_{2n}=\frac{x_{2n-1}+2x_{2n-2}}{3}\,\,\,\,\,,x_{2n+1}=\frac{2x_{2n}+x_{2n-1}}{3}$$ for $n \in \mathbb{N^+}$. Show that the sequence $(x_n)$ converges. my att... | Closed forms for linear recurrence relations can be found using matrices and diagonalization.
Let $v_i = \begin{bmatrix}x_i & x_{i - 1}\end{bmatrix}^\intercal$. We have $v_1 = \begin{bmatrix}b & a\end{bmatrix}^\intercal$ and for any positive integer $n$,
$$\begin{align}
v_{2n} &= \begin{bmatrix}1/3 & 2/3\\1 & 0\end{bma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/298768",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
A log improper integral Evaluate :
$$\int_0^{\frac{\pi}{2}}\ln ^2\left(\cos ^2x\right)\text{d}x$$
I found it can be simplified to
$$\int_0^{\frac{\pi}{2}}4\ln ^2\left(\cos x\right)\text{d}x$$
I found the exact value in the table of integrals:
$$2\pi\left(\ln ^22+\frac{\pi ^2}{12}\right)$$
Anyone knows how to evaluate t... | $$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { \sin }^{ 2m-1 }\left( x \right) { \cos }^{ 2n-1 }\left( x \right) dx } =B\left( m,n \right) \\ \int _{ 0 }^{ \frac { \pi }{ 2 } }{ { \sin }^{ 2m-1 }\left( x \right) { \left( { \cos }^{ 2 }x \right) }^{ \frac { 2n-1 }{ 2 } }dx } =B\left( m,n \right) $$
On differentiating it ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/300061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 5,
"answer_id": 1
} |
Rolling three dice...am I doing this correctly? Tree dice are thrown. What is the probability the same number appears on exactly two of the three dice?
Since you need exactly two to be the same, there are three possibilities:
1. First and second, not third
2. First and third, not second
3. Second and third, not first
F... | One more way you can solve this is using complimentary counting. There are 6*$\dbinom{6}{4}$=120 ways to get all numbers different. There are 6 ways to get all numbers same. So there are 216-126=90 ways that work. $\frac{90}{216}=\frac{5}{12}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/300965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 2
} |
Diophantine equations - Perfect square and Perfect cube related Solve following Diophantine equations:
$1) \ a^3-a^2+8=b^2$
2) $a, \ b,\ c \in \mathbb{Z^+}$$$\frac{a^3}{(b+3)(c+3)} + \frac{b^3}{(c+3)(a+3)} + \frac{c^3}{(a+3)(b+3)} = 7$$
3) $a^3-8=b^2$
In Problem 2 I tried to use inequality, then I can 'limit' that: $2... | (2) Extremely ugly solution.
You have
$$a^3(a+3)+b^3(b+3)+c^3(c+3)=7(a+3)(b+3)(c+3)$$
or
$$(x-3)^3x+(y-3)^3y+(z-3)^3z=7xyz$$
Since it is symmetric, we can look for the solutions where $x \geq y \geq z$.
It is easy to check that for $x>15$ we have $(x-3)^3>7x^2$. This shows that $4 \leq x \leq 14$. For each particular $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/301244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Assume $n$ is even. Prove that $323$ divides $20^n+16^n-3^n-1$. I'm unclear what is the best method to teach this with minimum math experience.
| We have $323 = 17 \cdot 19$ with $\gcd(17, 19) = 1$. Also, $19 = 20-1 \mid 20^n - 1, 19 = 16 + 3 \mid 16^n - 3^n \Rightarrow 19 \mid 20^n+16^n-3^n-1$
and $17 = 20 - 3 \mid 20^n - 3^n$, $17 = 16+1 \mid 16^n +1 \Rightarrow 17 \mid 20^n+16^n-3^n-1$
$\gcd(17,19)=1 \Rightarrow 323 = 17 \cdot 19 \mid 20^n+16^n-3^n-1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/301867",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
2nd order implicit derivative What would the 2nd order implicit derivative of $y^2=12x$ be?
I get the first derivative is $2yy'=12$ but Wolfram gives the second as $y''=\frac{-3}{xy}$ and I don't understand how they get that.
| $$ y^2 = 12x \Rightarrow \\ 2\cdot y \frac{dy}{dx} = 12 \Rightarrow \\ \frac{dy}{dx} = \frac{12}{2y} = \frac{6}{y} \Rightarrow \\ \frac{d^2y}{dx^2} = -\frac{6}{1}\cdot \frac{1}{y^2}\cdot \frac{dy}{dx} = -\frac{6}{1}\cdot \frac{1}{y^2}\cdot \frac{6}{y} = -\frac{36}{y^3} = -36\cdot \frac{1}{y^2\cdot y} = -36\cdot \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/302448",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Integer solution $(x,y,z)$ to $xyz=24$
Find the total number of integer solutions $(x,y,z)\in\mathbb{Z}$ of the equation $xyz=24$.
I have tried $xyz = 2^3 \cdot 3$
My Process:
Factor $x$, $y$, and $z$ as
$$
\begin{cases}
x = 2^{x_1} \cdot 3^{y_1}\\
y = 2^{x_2} \cdot 3^{y_2}\\
z = 2^{x_3} \cdot 3^{y_3}
\end{cases}
$$
... | Yes, this looks right for finding the number of positive integer solutions.
Since you originally asked for all integer solutions, you'll also have to take into account the signs of the factors. There are $\binom{3}{0} + \binom{3}{2} = 4$ ways of adding zero or two minus signs, for a total of 120 integer solutions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/307248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 0
} |
Finding the value of the integral $\int_{1/2}^{1}[1/x^2]dx$ I came across the following problem that says:
For any $x \in \mathbb R$,let $[x]$ denote the greatest integer smaller than or equal to $x$. Then the value of the integral $\int_{1/2}^{1}[1/x^2]dx$ is which of the following:
1.$\frac {1}{\sqrt 3}+\frac {... | When $x=1/2$, $[1/x^2] = 4$; when $x=1/\sqrt{3}$, it is equal to $3$, etc. The integral is then
$$\int_{1/2}^1 dx \: [1/x^2] = -4 \left ( \frac{1}{2} - \frac{1}{\sqrt{3}} \right ) - 3 \left ( \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{2}} \right ) - 2 \left ( \frac{1}{\sqrt{2}} - 1 \right )$$
or,
$$\int_{1/2}^1 dx \: [1/x^2]... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/307296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Algebraically Solve Limit $$\lim_{x \to 0} \dfrac{2\sqrt{x+1}-x-2}{x^2}$$
I can solve it using l'Hôpital but just cannot find a way to do it algebraically.
| We have $\sqrt{1+x}=1+\frac{1}{2}x+(\frac{1}{2})(\frac{-1}{2})\frac{x^2}{2}+o(x^2)$, so
$$\lim_{x \to 0} \dfrac{2\sqrt{x+1}-x-2}{x^2}=\lim_{x \to 0}\dfrac{-\frac{1}{4}x^2+o(x^2)}{x^2}=-\frac{1}{4}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/307456",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Show that the difference of two consecutive cubes is never divisible by $3$. Here is my proof:
Let $n \in \Bbb Z$. Then, $n$ is of the form $2k$(even) or $2k + 1$(odd), for some $k \in \Bbb Z$.
Without loss of generality (not sure if I can use this), let $n = 2k$.
Then, $n + 1 = 2k + 1$.
$$\begin{align}
(n + 1)^3 - n^... | $(n+1)^3 - n^3 = n^3 + 3n^2 + 3n + 1 - n^3 = 3(n^2 + n) + 1 \equiv 1$ (mod $3$)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/308449",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 1
} |
Proof by induction help. I seem to be stuck and my algebra is a little rusty Stuck on a homework question with mathematical induction, I just need some help factoring and am getting stuck.
$\displaystyle \sum_{1 \le j \le n} j^3 = \left[\frac{k(k+1)}{2}\right]^2$
The induction part is: $\displaystyle \left[\frac{k(k+1)... | You have:
$$
\begin{align*}
\left[ \frac{n (n + 1)}{2} \right]^2 + (n + 1)^3
&= \frac{n^2 (n + 1)^2 + 4 (n + 1)^2 (n + 1)}{4} \\
&= \frac{(n^2 + 4 (n + 1)) (n + 1)^2}{4} \\
&= \frac{(n^2 + 4 n + 4) (n + 1)^2}{4} \\
&= \frac{(n + 2)^2 (n + 1)^2}{4} \\
&= \left[\frac{(n + 1) (n + 2)}{2} \right]^2
\end{align*}
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/309570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Solving an equation with fractional powers I was trying to find the maximum value for a function. I took the first derivative and arrived at this horrible expression:
$$ (x^2 + y^2)^\frac{3}{2} - y {\frac{3}{2}}(x^2 + y^2)^{\frac{1}{2}}2y = 0$$
How can I find the extrema by hand?
| $$ (x^2 + y^2)^{3/2} - y {\frac{3}{2}}(x^2 + y^2)^{\frac{1}{2}}2y = 0$$
$$\iff (x^2 + y^2)^{1/2}\left((x^2 + y^2 - 3y^2)\right) = 0$$
$$\iff (x^2 + y^2)^{1/2}\left(x^2 - 2y^2\right) = 0$$
$$\iff (x^2 + y^2) = 0 \;\;\text{ or }\;\;x^2 = 2y^2$$
$$\iff \quad ?$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/311570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 2
} |
A improper integral with complex parameter For a complex number $\displaystyle z$, How to evaluate
$$\int_0^\infty\frac{\text{d}x}{x^2+(1-z^2x^2)^2}$$
| By the quadratic formula, the denominator of the integrand has roots
$$\begin{align}
r_1,r_2,r_3,r_4 &= \pm\sqrt{\frac{2z^2-1\pm\sqrt{1-4z^2}}{2z^4}}\\
&=\frac{\pm1}{2z^2}\left(i\pm\sqrt{4z^2-1}\right)
\end{align}$$
and since the integrand is an even function,
$$\begin{align}
\int_0^\infty \frac{dx}{x^2+(1-z^2x^2)^2} &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/311661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
What is the remainder when $x^{100} -8x^{99}+12x^{98}-3x^{10}+24x^{9}-36x^{8}+3x^{2}-29x + 41$ is divided by: $x^2-8x+12$. What is the remainder when: $$x^{100} -8x^{99}+12x^{98}-3x^{10}+24x^{9}-36x^{8}+3x^{2}-29x + 41$$ is divided by: $$x^2-8x+12$$
$x^2-8x+12$ $\leftrightarrow (x-2)(x-6)$
This gives me the polynomial:... | First note that $r(x)=ax+b$ for some $a,b\in\mathbb Z$
For $x=2$ you get $r(2)=-5$ (not $r(x)=-5$) $ \Rightarrow 2a+b=-5$. Do the same for $x=6$ to find r(6).Then find $a,b$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/313044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Is this function decreasing? I have to prove that the following function is decreasing when $x>0$
$$f(x)=\frac{(x+1)^2}{\sqrt{x^2+2x}}-(x+1)^2\arcsin\Big(\frac{1}{x+1}\Big)$$
but calculating its derivative I don't succed in proving this result.
| $f'(x) = \dfrac{3(x+1)-\dfrac{(x+1)^3}{(x+2)x}}{\sqrt{x(x+2)}}-2(x+1)\arcsin(\dfrac{1}{1+x})$
Since $1+x>0$,thus we just prove
$\dfrac{3-\dfrac{(1+x)^2}{(x+2)x}}{\sqrt{x(x+2}}-2\arcsin(\dfrac{1}{x+1})\le 0$
Since $\arcsin(\dfrac{1}{1+x})\ge \dfrac{1}{1+x}$, thus
we just prove
$\dfrac{3-\dfrac{(1+x)^2}{(x+2)x}}{\sqrt{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/315266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find parametric expression of an arc given its start point, end point and central angle in 3D cartesian coordinate system In a 3D cartesian coordinate system, the coordinates of start point and end point have been given as $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$. If the central angle of the two points (the one smaller ... | The expression for the locus of points of the circle containing the arc is
$$ {\rm RZ}(\psi) \begin{pmatrix}
x_c + r \cos \varphi \\
y_c \\
z_c + r \sin \varphi
\end{pmatrix} = \begin{pmatrix}
r \cos\varphi \cos\psi + x_c \cos\psi - y_c \sin\psi \\
r \cos\varphi \sin\psi + y_c \cos\psi + x_c \sin\psi \\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/316657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
$(y^2+2iy)^{m-\frac{1}{2}}=y^{m-\frac{1}{2}}(2i)^{m-\frac{1}{2}}+O(y^{m+\frac{1}{2}})(0\leq y \leq 1)?$ This question comes from stein's book Introduction to fourier analysis on euclidean space,page 159.
when $m > 1/2$,
$$(y^2+2iy)^{m-\frac{1}{2}}=y^{m-\frac{1}{2}}(2i)^{m-\frac{1}{2}}+O(y^{m+\frac{1}{2}})(0\leq y \leq ... | The first equality comes from the fact $\forall \alpha>0, (1+x)=_01+O(x)$ where $=_0$ means that the equality is true for $x$ in a neighborhood of $0$. Then we have:
$$(y^2+2iy)^{m-\frac{1}{2}}=(2iy)^{m-\frac{1}{2}}\left((\frac{1}{2i})^{m-\frac{1}{2}}y+1\right)=_0 (2iy)^{m-\frac{1}{2}}(1+O(y))\\=_0y^{m-\frac{1}{2}}(2i)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/317764",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
>Prove that $\frac d {dx} x^n=nx^{n-1}$ for all $n \in \mathbb R$.
Prove that $\frac d {dx} x^n=nx^{n-1}$ for all $n \in \mathbb R$.
I saw some proof of $\frac d {dx} x^n=nx^{n-1}$ using binomial theorem, which is only available for $n \in\mathbb N$. Do anyone have the proof of $\frac d {dx} x^n=nx^{n-1}$ for all rea... | From the definition of derivatives, we have
$$ f'(x) = \lim_{h \to 0}\frac{f(x + h) - f(x)}{h} $$
Let's assume here that $f(x) = x^n$. Then,
*
*Case 1, $n \in \mathbb{N}$
$$
\begin{align}
f'(x) &= \lim_{h \to 0} \frac{(x + h)^n - x^n}{h} \\
&= \lim_{h \to 0} \frac{x^n + nhx^{n-1} + \frac{n(n-1)}{2!}h^2x^{n-2} + \c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/318555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Show that the curve $x^2+y^2-3=0$ has no rational points
Show that the curve $x^2+y^2-3=0$ has no rational points, that is, no points $(x,y)$ with $x,y\in \mathbb{Q}$.
Update: Thanks for all the input! I've done my best to incorporate your suggestions and write up the proof. My explanation of why $\gcd(a,b,q)=1$ is a... | Suppose to the contrary that there is a rational solution of the equation. Then there exist integers $a$, $b$, and $q$, with $q\ne 0$, such that $a^2+b^2=3q^2$, and $a$, $b$, and $q$ have no common factor greater than $1$.
Note that $a$ and $b$ must both be divisible by $3$. For if an integer $m$ is not divisible by $3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/319553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 4,
"answer_id": 0
} |
$y''=y$, $x''=-x$. Write this equation in terms of the first-order system I'm having trouble with my homework on higher-order equations and their equivalent systems. :(
This is the problem:
Write this equation in terms of the first-order system.
$$\left\{ \begin{array}{l} \frac{d^2x}{dt^2}=-x \\ \frac{d^2y}{dt^2}=y \en... | If $X= \left( \begin{array}{c} x' \\ x \end{array} \right)$, then $X'= \left( \begin{array}{c} x'' \\ x' \end{array} \right) = \left( \begin{array}{c} -x \\ x' \end{array} \right)= \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right) \left( \begin{array}{c} x' \\ x \end{array} \right)=AX$. You can do the same t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/321312",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Solutions of the Diophantine equation $x^2(x^2+10)=3y^2(y^2+10)$ I am looking for the solutions of the Diophantine equation
$$x^2(x^2+10)=3y^2(y^2+10).$$
Is there any solution of this equation except when $(x,y)=(0,0)$?
Or
Any computer programme such as MAGMA could solve this problem?
Thank you very much.
| If $x$ and $y$ are not both $0$, let $3^k$ be the highest power of $3$ that divides both $x$ and $y$. Let $x=3^ks$ and $y=3^k t$. Then $3$ cannot be a common divisor of $s$ and $t$.
Substitute and cancel. We get $s^2(x^2+10)=3t^2(y^2+10)$. Since $3$ cannot divide $x^2+10$, it must divide $s$. Say $s=3u$. Then $9u^2(x^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/322444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Integrating $\int^0_\pi \frac{x \sin x}{1+\cos^2 x}$ Could someone help with the following integration:
$$\int^0_\pi \frac{x \sin x}{1+\cos^2 x}$$
So far I have done the following, but I am stuck:
I denoted $ y=-\cos x $ then:
$$\begin{align*}&\int^{1}_{-1} \frac{\arccos(-y) \sin x}{1+y^2}\frac{\mathrm dy}{\sin x}\\&... | Let $$I = \int_0^{\pi} \dfrac{x \sin(x)}{1+\cos^2(x)} dx = \int_{-\pi/2}^{\pi/2} \dfrac{(x+\pi/2) \sin(x+\pi/2)}{1 + \cos^2(x+\pi/2)} dx = \int_{-\pi/2}^{\pi/2} \dfrac{(x+\pi/2) \cos(x)}{1 + \sin^2(x)} dx $$
Now
$$\int_{-\pi/2}^{\pi/2} \dfrac{(x+\pi/2) \cos(x)}{1 + \sin^2(x)} dx = \int_{-\pi/2}^{\pi/2} \underbrace{\dfr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/323109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 0
} |
Fourier series of function $f(x)=0$ if $-\pi$$f(x) = \begin{cases}0 & \text{if }-\pi<x<0, \\
\sin(x) & \text{if }0<x<\pi.
\end{cases}$$
My attempt:
I went the route of expanding this function with a complex Fourier series.
$$f(x) = \sum_{n=-\infty}^{+\infty} C_{n}e^{inx}$$
$$C_{n} = \frac {1}{2\pi} \int_{0}^{\pi} \frac... | The $\sin$ term comes form $n=1$ you can't devide by zero.
mh I calculated again, your $\cos(x)$ terms are right, there shouldn't be a $-$ in the denominator
For $$\int_0^\pi \sin(x) e^{inx}\, \mathrm{d} x = \frac{1+ e^{i \pi n}}{1-n^2}$$
we have to check the case $n=1$ seperate as we can't devide by zero.
The case $n=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/324073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 0
} |
Partial Derivative of $f(x,y) =\ln(x^{2} + y^{2}) + \sqrt{x^{2}\cdot y^{3}}$ $$f(x,y) = \ln(x^{2} + y^{2}) + \sqrt{x^{2}\cdot y^{3}}$$
What is the value of $f_{x}\left ( 0,1 \right )$ and $f_{y}\left ( 0,1 \right )$?
I tried but I found the denominator as zero.
| $$f(x,y) = \ln(x^{2} + y^{2}) + \sqrt{x^{2}\cdot y^{3}}$$
Using chain rule of differentiation:
$$
\begin{align}
f_x(x, y) &= \dfrac{2x}{x^2 + y^2} + \dfrac{1}{2} 2xy^3 \left(x^2 \cdot y^3\right)^{\frac{-1}{2}} \\
&= \dfrac{2x}{x^2 + y^2} + \dfrac{xy^3}{\sqrt{x^2 \cdot y^3}} \\
&= \dfrac{2x}{x^2 + y^2} + y^{\frac{3}{2}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/324980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Variable Exponents Question If $a,b$ and $c$ are different positive integers and $2^a\cdot2^b\cdot2^c =64$ then $2^a+2^b+2^c$=?
This is so far my work: I got $2^a\cdot2^b\cdot2^c=2^6$ then $abc=6$ is this so far in the right track?
| $$2^a \cdot 2^b\cdot 2^c = 2^{a + b + c} = 2^6\;\;\iff\;\; a + b + c = 6$$
(Recall that $2^x\cdot 2^y = 2^{x+y}.$)
The only possible combinations of distinct $a, b, c$ which sum to $6$ is $\;(a, b, c) = (1, 2, 3),\;$ or any permutation thereof.*
That is, $$a \neq b \neq c \implies 2^a + 2^b + 2^c = 2^1 + 2^2 + 2^3 = 2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/325127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
How to solve $\int\cot^5x\sin^2x\ dx$? I'm not quite sure how to approach this without it getting extremely messy... and even then, I don't know if it will come out right.
The best I can think of is to use IBP, but neither of those functions are easy to integrate.
I did integrate $\sin^2x$ to get $$\int\sin^2x\ dx=\fra... | We have:
\begin{align}
I = \int \cot^5 x \sin^2 x \,dx &= \int \frac{\cos^5 x}{\sin^5 x} \sin^2 x \,dx \\
&= \int \frac{\cos^5 x}{\sin^3 x} \,dx \\
&= \int \frac{\left(1 - \sin^2 x\right)^2}{\sin^3 x} \cos x \,dx
\end{align}
Now put $u = \sin x$ to get:
\begin{align}
I &= \int \frac{\left(1 - u^2\right)^2}{u^3} \,du \\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/326049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
$\overline{xyz} =x!+y!+z!$ how to find all of 3-digit integer number $\overline{xyz}$ such that:
$\overline{xyz} =x!+y!+z!$
| As I was partway through typing this answer, Dan Shved's answer appeared which is much better than mine, but I'll include what I have so far as I use some different ideas. I reduce the number of possibilities to $13$ (though of course, I could reduce this number even further using the reasons in Dan's answer).
\begin{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/327352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 2
} |
Solving: $\frac{3x-1}{2} =\frac{-2}{x+2}$ How to solve :
$$\frac{3x-1}{2} =\frac{-2}{x+2} $$
| $$\frac{3x-1}{2} =\frac{-2}{x+2} $$
$$(x+2)(3x-1)=2\cdot(-2) $$
$$3x^2+5x+2=0$$
$$a=3,b=5,c=2$$
$$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/327570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
System of equations wrt self-adjoint operators $X = \left( \begin{matrix} 2&s\\ 8&2 \end{matrix} \right)$ and $Y = \left( \begin{matrix} 2&-1\\ 2&2 \end{matrix} \right)$ are two operators wrt the same orthonormal basis $B$ in a 2D scalar product space. Furthermore, $s$ is a fixed, complex-valued parameter.
The associat... | We have $XY^\ast=P^3$. Therefore $XY^\ast=\begin{pmatrix}4-s&2s+4\\14&20\end{pmatrix}$ is self-adjoint and hence $s=5$. Note that $P,Q$ are invertible because $X,Y$ are invertible. Also, since $XQ=P^2$ and $QY^{-1}=P^{-1}$ are self-adjoint, we have
\begin{align}
XQ&=QX^T,\tag{1}\\
QY^{-1}&=Y^{-T}Q,\tag{2}\\
XQ &= (YQ^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/327790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Solve $y^2= x^3 − 33$ in integers This is not homework, could someone provide a nice clear proof as I have been struggling with this for some time.
Solve the equation $y^2= x^3 − 33$; $x, y \in \mathbb{Z}$
| $$ y^2 + 25 = x^3 - 8. $$
Note $y^2 + 25$ can not be divisible by any prime $q \equiv 3 \pmod 4,$ therefore not by any number $m \equiv 3 \pmod 4.$
If $y$ were odd, $y^2 + 25 \equiv 2 \pmod 4,$ impossible for $x^3 - 8.$ Therefore $y$ is even and $x$ is odd.
$$ y^2 + 25 = (x - 2)(x^2 + 2 x + 4). $$
As we need $x-2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/330779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
} |
How to prove this inequality? $ab+ac+ad+bc+bd+cd\le a+b+c+d+2abcd$ let $a,b,c,d\ge 0$,and $a^2+b^2+c^2+d^2=3$,prove that
$ab+ac+ad+bc+bd+cd\le a+b+c+d+2abcd$
I find this inequality are same as Crux 3059 Problem.
| Your constraints are $a,b,c,d > 0$ and $a^2+b^2+c^2+d^2 = 3$, by doing a change of variables, $x^2 = a^2/3, y^2 = b^2/3, z^2 = c^2/3, w^2 = d^2/3$. Then your constraints change to
$$x,y,z,w > 0, x^2+y^2+z^2+w^2 =1$$
and the problem is equivalent to proving $xy +xz +xw +yz+yw+zw \leq \sqrt{3}(x+y+z+w) + \frac{2}{3}xyzw... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/331004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 3,
"answer_id": 1
} |
Generating function for the sequence $1,1,3,3,5,5,7,7,9,9,\ldots$ The generating function for the sequence $\left\{1,1,1,1,...\right\}$ is $$1 + x + x^2 + x^3 ... = \frac{1}{1-x}$$
What is the generating function for the sequence $\left\{1,1,3,3,5,5,7,7,9,9,\dots \right\}$?
This is my attempt:
what we want to do is af... | You already did the difficult part... You probably mean $$\frac1{1-x}+\sum_{k=1}^\infty\frac{2x^{2k}}{1-x}$$
So you just have to evaluate the geometric series $$\sum_{k=1}^\infty (x^2)^k = \frac1{1-x^2} - 1 = \frac{x^2}{1-x^2}$$
And obtain
$$\frac1{1-x} + \frac2{1-x}\sum_{k=1}^\infty x^{2k} = \frac{1+\frac{2x^2}{1-x^2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/331058",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Integral $\int\frac{6^x}{9^x-4^x}dx $ How to solve this integral:
$$\int\frac{6^x}{9^x-4^x}dx $$
(I notice that $\frac{6^x}{9^x-4^x}=\frac{2^x3^x}{(3^x-2^x)(3^x+2^x)}$)
Thank you!
| Note that $$\frac{1}4\cdot\frac{3^x+2^x}{3^x-2^x}-\frac{1}4\cdot\frac{3^x-2^x}{3^x+2^x}$$ is the integrand. Now use what @J.H. suggested.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/331499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 2
} |
$ \sum_{n=2}^\infty \frac{1}{n^3(n^3+1)}. $ The series is:
$$
\sum_{n=2}^\infty \frac{1}{n^3(n^3+1)}.
$$
I tried splitting the whole thing into simple fractions but I don't seem to get anywhere.
Any ideas?
| $$\sum_{n=2}^\infty \frac{1}{n^3(n^3+1)} = \sum_{n=2}^\infty \frac{1}{n^3}+\frac{n-2}{3 (n^2-n+1)}-\frac{1}{3 (n+1)} = \\
\zeta{(3)} -1 + \sum_{n=2}^{\infty} \frac{n-2}{3 (n^2-n+1)}-\frac{1}{3 (n+1)} = \\
\zeta{(3)}-1 -\sum_{n=2}^{\infty} \frac{1}{n^3+1} \approx \\
0.015553560820970399435132949324595238582753593120862... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/331850",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
What are the generators for $SL_n(\mathbb{R})$ (Michael Artin's Algebra book) The book asks you to prove that $SL_n(\mathbb{R})$ is generated by elementary (row operation) matrices in which one nonzero off-diagonal entry is added to the identity matrix. For example,
$$
\begin{bmatrix}
1 & a \\
0 & 1
\end{bmatrix}
$$
... | The point is, you can do without using elementary row operations of the form $R_i\leftarrow \lambda R_i$. Given a matrix $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ such that its first column is nonzero, we can always reduce its first column to $(1,0)^T$ using only elementary row operations of the form $R_i\leftarrow R_i+kR... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/333496",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Tensor and Kronecker product I have a follow exercise. Let $\left|\Psi\right\rangle = \tfrac{1}{\sqrt{2}}\left(\left|0\right\rangle + \left|1\right\rangle\right)$. Write out $\left|\Psi\right\rangle^{\otimes 2}$ xplicitly, both
in terms of tensor products like $\left|0\right\rangle \otimes \left|1\right\rangle$ , and u... | Let $S : \mathbb{C}^2 \otimes \mathbb{C}^2 \to \mathbb{C}^4$ be the isomorphism that maps the tensor product $v \otimes w$ of $v$, $w \in \mathbb{C}^2$ to their Kronecker product. Then, in general, for $v = (v_1,v_2)^T$, $w = (w_1,w_2)^T \in \mathbb{C}^2$,
$$
S\left(v \otimes w\right) = \begin{pmatrix} v_1 w \\ v_2 w ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/334168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How many solutions are possible to this equation? Given
$$A+2B+3C=N
$$
where $N$ is a given positive integer.
$A ,B,C\in\mathbb{N}$ vary from $0$ to $\infty$.
How many solutions will be there to this equation?
| Let $X=C, Y=B+C, Z=A+B+C$, then this is equivalent to finding the number of non-negative integer solutions to $X+Y+Z=N$ s.t. $X \leq Y \leq Z$.
First ignore the restriction on $X, Y, Z$. In total we have $\binom{N+2}{2}$ solutions.
The number of solutions with $Y=Z$ is just the number of non-negative integer solution... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/334901",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
$\left(\frac{\sin(\frac{n\theta}{2})}{\sin(\frac{\theta}{2})}\right)^2=\left|\sum_{k=1}^{|n|}e^{ik\theta}\right|^2$ I'm having trouble proving $$\left(\frac{\sin(\frac{n\theta}{2})}{\sin(\frac{\theta}{2})}\right)^2=\left|\sum_{k=1}^{|n|}e^{ik\theta}\right|^2$$ where $n\in\mathbb{Z}$ and $\theta\in\mathbb{R}$. Can anyon... | Using the sum of sines and cosines with arguments in arithmetic progression as given above: if $\theta\ne0$ and let $\varphi =0$, then we have,
\begin{align} &S =\sin{(\theta)} + \sin{(2\theta)} + \cdots + \sin{(n\theta)} = \frac{\sin{\left(\frac{(n+1) \theta}{2}\right)} \cdot \sin{(\frac{n \theta}{2})}}{\sin{\frac{\th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/335651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Establish convergence of the series: $1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-...$ Establish convergence of the series: $1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-...$
The number of signs increases by one in each "block".
I have an idea. Group the series like this: $1-(\frac{1}... | If I'm not getting it wrong, your series is
$$1-\frac 1 2 -\frac 13+\frac 14+\frac 15+\frac 16-\frac 17-\frac 18-\frac 19-\frac 1{10}+++++------\dots$$
So you have $1$ plus, $2$ minuses, $3$ pluses, $4$ minuses, and so on.
We can write your series as $a_0+a_1+a_2+a_3+\dots$ where
$$\begin{align} a_0&=(-1)^0 1 \\
a_1&=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/336035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 6,
"answer_id": 0
} |
Find all positive integers solution of $xy+yz+xz = xyz+2$ 1.Find all positive integers solution
$xy+yz+xz = xyz+2$
2.Determine all p and q which p,q are prime number and satisfy
$p^3-q^5 = (p+q)^2$
Thx for the answer
3.Find all both positive or negative integers that satisfy
$\frac{13}{x^2} + \frac{1996}{y^2} = \frac{... | 1.set $x≤y≤z,xyz<xyz+2=xy+yz+zx≤3yz,$
so $0<x<3,x=1,2$
If $x=1,y+z+yz=yz+2,y+z=2$,so $x=y=z=1$
If $x=2,2y+2z+yz=2yz+2,yz=2y+2z-2<4z,2≤y<4,y=3,z=4$
so the only solution to the fisrt equation is $x=y=z=1$ or $x=2,y=3,z=4$ or change their order.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/336556",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Write in polynomial in factored form in complex number Write the following polynomial in factored form(in complex number):
$$1+z+z^2+z^3+z^4+z^5+z^6$$
Also, is there general solution of factoring for $1+z+z^2...z^n$ types of polynomial?
| I'd extend Adi Dani's answer a bit
$$
1 + z^2 + \ldots + z^6 = \frac {1-z^7}{1-z}
$$
Now decompose $1-z^7$ in factors. In order to do that, find all roots of
$$
1-z^7 = 0 \\
z = \sqrt [7]1 = \sqrt[7]{e^{0\cdot pi}} = \{ e^{\frac {2\pi ki}7}\}, k = 0,1,\ldots,6
$$
So
$$
1-z^7 = \prod_{k = 0}^6 \left (z - e^{\frac {2\pi ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/337473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
solving $x^2 = a \pmod {2^n}$ , $n \ge 3$ I read that the equation $x^2 = a \pmod {2^n}$ for $n \ge 3$ has four solutions and the solutions are $x_1, -x_1, x_1 + 2^{n-1}, - x_1 + 2^{n-1}$. It is easy to prove that they are indeed the solutions and are incongruent solutions. But if I need to derive that these are the on... | We need to assume that $a$ is odd. For example, if $a=4$, modulo $32$ we have solutions $2,6,10,14,18,22,26,30$.
Suppose that $x_1^2\equiv a \pmod{2^n}$ and $x\equiv a\pmod{2^n}$. We need to show that $x$ can take on at most $4$ values.
From the two congruences, we have $x^2\equiv x_1^2\pmod{2^n}$, meaning that
$2^n$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/338160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to find $\lim_{n\to\infty}n^2\left(\sin(2\pi en!)-\frac{2\pi}{n}\right)$
How to find$$\lim_{n\to\infty}n^2\left(\sin(2\pi en!)-\frac{2\pi}{n}\right)$$
I am very confused. I don't know how to show the limit exists, or what it is.
| Recall that $e = 1 + \dfrac1{1!} + \dfrac1{2!} + \dfrac1{3!} + \cdots + \dfrac1{n!} + \dfrac1{(n+1)!} + \dfrac1{(n+2)!} + \cdots$. Hence,
$$n! e = \text{Integer} + \dfrac1{n+1} + \dfrac1{(n+1)(n+2)} + \dfrac1{(n+1)(n+2)(n+3)} + \cdots$$
Hence,
\begin{align}
\sin(2\pi n!e) & = \sin\left(\text{Integer} \times 2 \pi + \df... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/339315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Surface area of an elliptic paraboloid The elliptic paraboloid has height h, and two semiaxes a, b. How to find its surface area? Does it possible to use a direct formula without integrals?
| The elliptic paraboloid is represented parametrically as follows:
$$x=a \sqrt{u} \cos{v}$$
$$y=b \sqrt{u} \sin{v}$$
$$z=u$$
The surface area of this object is given by
$$\int_0^h du \: \int_0^{2 \pi} dv \: \sqrt{E \,G - F^2}$$
where the 1st fundamental form is given by
$$E=1+\frac{a^2 \cos^2{v} + b^2 \sin^2{v}}{4 u}$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/339621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Finding determinant (using row reduction) I'm trying to calculate the following
$\det \begin{pmatrix} x-1 & -1 & -1 & -1 & -1\\ -1 & x-1 & -1 & -1 & -1 \\ -1 & -1 & x-1 & -1 & -1 \\ -1 & -1 &-1 & x-1 & -1 \\ -1 & -1 & -1 & -1 & x-1 \end{pmatrix}$
and I'm sure there must be a way to get this as an upper/lower triangular... | By Gaussian elimination, using the the first entry of the fifth row as the first pivot, we get that
$$
\begin{vmatrix} -1+x & -1 & -1 & -1 & -1 \\ -1 & -1+x & -1 & -1 & -1 \\ -1 & -1 & -1+x & -1 & -1 \\ -1 & -1 & -1 & -1+x & -1 \\ -1 & -1 & -1 & -1 & -1+x \end{vmatrix}
= \begin{vmatrix} -1 & -1 & -1 & -1 & -1+x \\ -1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/340179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to prove ${{a}^{a}}{{b}^{b}}\ge {{\left(\frac{a+b}{2}\right)}^{a+b}}$ ?thanks. How to prove
$${{a}^{a}}{{b}^{b}}\ge {{\left(\frac{a+b}{2}\right)}^{a+b}}$$
$a>0$,$b>0$,
thanks.
| Dividing by $b^{a+b}$ and taking the $b^{th}$ root, we need to prove
$$\left(\dfrac{a}{b} \right)^{a/b} \geq \left(\dfrac{a/b+1}2 \right)^{a/b+1}$$
Let $a/b = t$. We then need to prove that
$$t^t \geq \left(\dfrac{1+t}2\right)^{1+t}$$
Consider the function
$$f(x) = x \log(x) - (1+x) \log \left(\dfrac{1+x}2 \right)$$
We... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/341439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 8,
"answer_id": 0
} |
What is the first non zero digit in 50 factorial (50!)? What is the first non zero digit in 50 factorial (50!)?
Any Help or hint will be appreciated.
| First, the exponent of $2$ in $50!$ is $47$, and the exponent of $5$ in $50!$ is $12$, so we’re looking for $\frac{50!}{10^{12}}$ mod$ 10$.
Since $\frac{50!}{10^{12}}$ is even (in fact, divisible by $2^{35}$), it suffices to compute it mod $5$:
$\frac{50!}{5^{12}}$ $\equiv$ ($4!$. $1$. $4!$. $2$. $4!$. $3$. $4!$. $4$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/341578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Fibonacci and the algebraic expression $x^2-x-1$ $$\left( \left( 1/2\,{\frac {-\beta+ \sqrt{{\beta}^{2}-4\,\delta\,
\alpha}}{\alpha}} \right) ^{i}- \left( -1/2\,{\frac {\beta+ \sqrt{{
\beta}^{2}-4\,\delta\,\alpha}}{\alpha}} \right) ^{i} \right) \left(
\sqrt{{\beta}^{2}-4\,\delta\,\alpha} \right) ^{-1}$$
What exactl... | Let $F_0=0$, $F_1=1$, $\forall n \in \Bbb N, F_{n+2}=F_{n+1}+F_n$
It's the Fibonacci sequence.
Now let $\forall n \in \Bbb N,V_n=\begin{pmatrix}
F_n\\F_{n+1}\end{pmatrix}$
Let $M=\begin{pmatrix}0 & 1\\ 1 & 1\end{pmatrix}$
Then $MV_n=\begin{pmatrix}0 & 1\\ 1 & 1\end{pmatrix}\begin{pmatrix}
F_n\\F_{n+1}\end{pmatrix}=\beg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/342096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
$\sin{\frac{A+B}{2}}+\sin{\frac{B+C}{2}}+\sin{\frac{C+A}{2}} > \sin{A}+\sin{B}+\sin{C}. $ Help me please to prove that:
for any $\triangle ABC$ we have the following inequality: $$\sin{\frac{A+B}{2}}+\sin{\frac{B+C}{2}}+\sin{\frac{C+A}{2}} > \sin{A}+\sin{B}+\sin{C}. $$
It's about convexity ?
thanks :)
| $\sin A+\sin B=\sin(\frac{A+B}{2}+\frac{A-B}{2})+\sin(\frac{A+B}{2}-\frac{A-B}{2})=2\sin\frac{A+B}{2}\cos\frac{A-B}{2}\leqslant2\sin\frac{A+B}{2}$. Similarly, $\sin B+\sin C\leqslant2\sin\frac{B+C}{2},\sin C+\sin A\leqslant2\sin\frac{A+C}{2}$. Add them together yields the result.
Note that the inequality holds is due t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/342597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
The least possible value How to find the least possible value for :$$(x-1)^2+(x-2)^2+(x-3)^2+(x-4)^2+(x-5)^2$$
For every real $x$
| Hint:
$(x-1)^2+(x-2)^2+(x-3)^2+(x-4)^2+(x-5)^2 \ge 0$
$5x^2+1+4+9+16+25 \ge 15x$
$5x^2 +55 -15x \ge 0 \implies x^2+11-3x \ge 0$
Aliter: $(x-3)=k$
You get
$(k+2)^2+(k+1)^2+(k)^2+(k-1)^2+(k-2)^2 \ge 0$
$k^2+11 \ge 0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/343841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Show that $\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x} = \frac{1}{2}\left(\tan 27x-\tan x\right)$ Show that
$$\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x} = \frac{1}{2}\left(\tan 27x-\tan x\right)$$
| A more general result is in fact true.
$$\sum_{k=0}^{n} \dfrac{\sin\left(3^k x\right)}{\cos\left(3^{k+1} x\right)} = \dfrac{\tan\left(3^{n+1} x\right) - \tan(x)}2 \,\,\,\, (\spadesuit)$$
Take $n=2$ in $(\spadesuit)$, to get what you want.
To prove $(\spadesuit)$, first note that
$$\dfrac{\sin \left(3^k \cdot x \right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/344505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Prove that $\sum_{k=n^2+1}^{n^2+2n+1}\sqrt{k}\le 2n^2+2n+\frac{7}{6}$ prove that:
$$\displaystyle\sum_{k=n^2+1}^{n^2+2n+1}\sqrt{k}\le 2n^2+2n+\dfrac{7}{6},n\ge 1$$
I find this inequality is very strong. Thank you!
such:when $n=100$,we use the mathmatic
$$\displaystyle\sum_{k=10001}^{10201}\approx 20201.16666254125412... | Note that your bound gets much tighter as $n$ increases, suggesting that some kind of limiting argument is involved.
With that insight, we have that
$$ \begin{align}
\sum_{k = n^2 + 1}^{k=n^2+2n+1} \sqrt{k} & = \sum_{k=n^2} ^{k=n^2 + 2n} \sqrt{k} + 1 \\
& = 1 + \sum_{k=n^2} ^{k=n^2+2n} \int_k^{k+1} \sqrt{k} \, dx \\ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/344776",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
$f(x - 1) + f(x − 2) $ and the sum of coeficients If $f(x-1)+f(x-2) = 5x^2 - 2x + 9$
and
$f(x)= ax^2 + bx + c$
what would be the value of $a+b+c$?
I was doing
$f(x-1)+f(x-2)= f(x-3)$
then
$f(x)$
a = 5
b = -2
c = 9
$(5-3)+(-2-3)+(9-3)$
But do not think is is correct
What would be correct approach?
| If $f(x) = ax^2 + bx+ c$, what is $f(x-1)$ and what is $f(x-2)$?
Work those out, then add both expressions to equate to $5x^2-2x+9$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/345094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Multivariable limit $\lim_{(x,y)\to (0,0)} \frac {x^2 + y^2}{\sqrt{x^2 +y^2 + 1} - 1}$ $$ \lim \limits_{(x,y)\to (0,0)} \frac {x^2 + y^2}{\sqrt{x^2 +y^2 + 1} - 1} $$
According to my textbook the limit equals $2$.
What I have tried:
Using the squeeze theorem:
$$ \lim \limits_{(x,y)\to (0,0)} \frac {x^2 + y^2}{\sqrt{x^2 ... | Remember the difference of squares algebraic identity.
$$ A^2 - B^2 = (A - B)(A + B) $$
Why is that useful? With $A = \sqrt{x^2 + y^2 + 1}$ and $B = 1$, the denominator of your expression is $A - B$. With that in mind,
$$ \begin{align}
\frac{x^2 + y^2}{\sqrt{x^2 + y^2 + 1} - 1} &= \frac{x^2 + y^2}{\sqrt{x^2 + y^2 + 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/345262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
What goes wrong in this derivative? $$ f(x) = \frac{2}{3} x (x^2-1)^{-2/3} $$
and f'(x) is searched.
So, by applying the product rule $ (uv)' = u'v + uv' $ with $ u=(x^2-1)^{-2/3} $ and $ v=\frac{2}{3} x $, so $ u'=-\frac{4}{3} x (x^2-1)^{-5/3} $ and $ v' = \frac{2}{3} $, I obtain
$$ f'(x) = - \frac{2}{9} (x^2-1)^{-5/3... | Your $u'v$ is wrong. Check your calculation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/346797",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
Cubes, squares and minimal sums I have trouble solving the following task: i need to find positive integers a and b such that
1) $a \neq b$
2) $ \exists c \in \mathbb{N} : ~ a^2 + b^2 = c^3$
3) $\exists d \in \mathbb{N}: ~ a^3 + b^3 = d^2$
4) sum $a + b$ is minimal possible
Thanks in advance!
| If $a=1250$ and $b=625$, then $a=2\cdot 5^4$ and $b=5^4$, and
$$
a^2 + b^2 = 2^2\cdot 5^8 + 5^8 = (4 + 1)5^8 = 5^9 = (5^3)^3
$$
$$
a^3 + b^3 = 2^3\cdot 5^{12} + 5^{12} = (8 + 1)5^{12} = 3^2(5^6)^2 = (3\cdot 5^6)^2
$$
This was the smallest example I found by an extremely primitive computer search.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/347624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove $\frac{1}{2a+2bc+1} + \frac{1}{2b+2ca+1} + \frac{1}{2c+2ab+1} \ge 1$ If $a,b$ and $c \ge 0$ and $ab + bc + ca = 1$, prove that the following inequality holds:
$$\frac{1}{2a+2bc+1} + \frac{1}{2b+2ca+1} + \frac{1}{2c+2ab+1} \ge 1$$
I've tried two aproaches, but it seems like both doesn't work. Here they are:
Cauch... | Let $a+b+c=3u$, $ab+ac+bc=3v^2$ and $abc=w^3$.
Hence, our inequality it's $f(w^3)\geq0$, where $f(w^3)=-8w^6+A(u,v^2)w^3+B(u,v^2)$.
But $f$ is a concave function, which says that it's enough to prove our inequality for an extremal value of $w^3$ wich happens in the following cases.
*
*$w^3=0$.
Let $c=0$. Hence, $a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/349577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 1
} |
Prove that for every positive integer $n$, $1/1^2+1/2^2+1/3^2+\cdots+1/n^2\le2-1/n$ Base case: n=1. $1/1\le 2-1/1$. So the base case holds.
Let $n=k\ge1$ and assume
$$1/1^2+1/2^2+1/3^2+\cdots+1/k^2\le 2-1/k$$
We want to prove this for $k+1$, i.e.
$$(1/1^2+1/2^2+1/3^2+\cdots+1/k^2)+1/(k+1)^2\le 2-\frac{1}{k+1}$$
This i... | $\displaystyle \frac{1}{i^2}< \frac{1}{i(i-1)}=\frac{1}{i-1}-\frac{1}{i}$,for $i\ge 2$
So we have $\displaystyle 1+\sum_{i=2}^{k}\frac{1}{i^2}\le 1+\sum_{i=2}^{k}(\frac{1}{i-1}-\frac{1}{i})=2-\frac{1}{k}$
I think this is better than induction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/351166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Show that $x^4-10x^2+1$ is reducible over $\mathbb{Q}(\sqrt{6}), \mathbb{Q}(\sqrt{2}),$ and $\mathbb{Q}(\sqrt{3})$. Is there an easier method? Ok, I actually worked this out as I was typing it up. But my solution seems kind of inelegant and involves a lot of tedious algebra that I've omitted here. Can anyone think of a... | Well, a possibility is to write it as a difference of two squares and thus factors into a sum and difference. For e.g.:
$$x^4-10x^2+1 = (x^2-5)^2 - 24$$
and noting that $24=4\cdot6$ is a square in $\mathbb{Q}(\sqrt{6})$, so this is the difference of two squares.
Similarly, $x^4-10x^2+1 = (x^2-1)^2 - 8x^2$ and note $8... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/351303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Sum of all numbers x such that $(3x^2+9x-2012)^{(x^3-2012x^2-10x+1)} = 1$? What is the sum of all $x$ such that $(3x^2 + 9x - 2012)^{(x^3-2012x^2-10x+1)} = 1$?
| Recall that $a^0 = 1$ for all real $a \neq 0$, and $1^b = 1$ for all real $b$.
$$(3x^2 + 9x - 2012)^{\large(x^3-2012x^2-10x+1)} = 1 \iff$$
$$ $$
$$x^3 \color{blue}{\bf - 2012}x^2 - 10 x + 1 = 0\tag{1}$$
or $$3x^2 + 9x - 2012 = 1 \iff 3x^2 + 9x - 2013 = 0 \iff \color{red}{\bf 1}\cdot x^2 + \color{red}{\bf 3}x - 671 = 0\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/353362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Find the Laurent expansion in powers of $\,z\,$ and $\,1/z\,$ $f(z)=\large\frac{z+2}{z^2-z-2}$ in $1<|z|<2$ and then in $2<|z|<\infty$
I am unsure how the two different regions of $z$ affect the series expansion.
Any help would be appreciated.
| $$f(z) = \frac{z+2}{z^2-z-2} = \frac{z+2}{(z-2)(z+1)} = \frac{4}{3(z-2)} - \frac{1}{3(z+1)} = -\frac{4}6 \frac{1}{z(1-\frac{z}{2})} - \frac{1}3 \frac{1}{z(1-\frac{-1}{z})} = -\frac{2}{3z} \sum_{k=0}^{\infty}\frac{z^k}{2^k} - \frac{1}{3z} \sum_{k=0}^{\infty}\frac{(-1)^k}{z^k}$$
This should give you convergence inside th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/354345",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Finding the value of $\sum\limits_{k=0}^{\infty}\frac{2^{k}}{2^{2^{k}}+1}$ Does this weighted sum of reciprocals of Fermat numbers,
$$
F=\sum_{k=0}^{\infty}\dfrac{2^{k}}{2^{2^{k}}+1}
$$
have a nice closed form? Wolfram says it's $1$.
Thanks.
| This might be another way of looking at Jyrki's hint, but here is the way I did this:
$$
\begin{align}
\color{#C0C0C0}{1-\frac1{2-1}+}\frac1{2+1}&=1-\frac2{4-1}\\
1-\frac{2}{4-1}+\frac{2}{4+1}&=1-\frac{4}{16-1}\\
1-\frac{4}{16-1}+\frac{4}{16+1}&=1-\frac8{256-1}\\
1-\frac8{256-1}+\frac8{256+1}&=1-\frac{16}{65536-1}\\
&\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/354796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 3,
"answer_id": 2
} |
Generating Functions- Closed form of a sequence We are given the following generating function : $$G(x)=\frac{x}{1+x+x^2}$$
The question is to provide a closed formula for the sequence it determines.
I have no idea where to start. The denominator cannot be factored out as a product of two monomials with real coefficien... | If you want the Binet formula for this you can solve the matrix
Set up the partial fraction
$$\frac{x}{1+x+x^2}=\frac{A}{1-ax}+\frac{B}{1-bx}$$ where $$a=\frac{-1+\sqrt{3}i}{2}$$ and $$ b=\frac{-1-\sqrt{3}i}{2}$$ so that
$$A(1-bx)+B(1-ax)$$
The matrix becomes
\begin{bmatrix}
-b&-a&1\\[0.3em]
1&1&0
\end{bmatrix}
or
\b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/355158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
How can I prove that $xy\leq x^2+y^2$? How can I prove that $xy\leq x^2+y^2$ for all $x,y\in\mathbb{R}$ ?
| \begin{align}
0\leq (x-y)^2 \implies
&
0\leq x^2-2xy +y^2
\\
\implies
&
2xy\leq x^2+y^2
\\
\implies
&
xy\leq \dfrac{x^2+y^2}{2}
\\
\implies
&
xy\leq {x^2+y^2}
\end{align}
since the clearly nonnegative real $x^2+y^2$ clearly satisfies $\dfrac{x^2+y^2}{2} \leq x^2+y^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/357272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "38",
"answer_count": 25,
"answer_id": 9
} |
Solving functional equation for generating function Find the functional equation for the generating function whose coefficients satisfy $$
a_n = \sum_{i=1}^{n-1}2^ia_{n-i}, \text{ for } n\ge 2, a_0 = a_1 = 1
$$
This is what I've tried so far:
$$
\begin{align}
g(x) -1 -x &= \sum_{n\ge2} \sum_{i=1}^{n-1} 2^i a_{n-i}\\
&=... | Let
$$g(x) = \sum_{k=0}^{\infty} a_k x^k$$
From the defining recurrence:
$$\begin{align}a_2 x^2 &= 2 a_1 x^2\\ a_3 x^3 &= 2 a_2 x^3 + 4 a_1 x^3\\a_4 x^4 &= 2 a_3 x^4 + 4 a_2 x^4 + 8 a_1 x^4\\a_5 x^5 &= 2 a_4 x^5 + 4 a_3 x^5 + 8 a_2 x^5 + 16 a_1 x^5\\ \ldots \end{align}$$
and so on. We then form
$$g(x) - a_0 - a_1 x =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/357888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
If $ a+b+c = \frac{9}{2}$ and $a,b,c>0$, then what is the minimum value of $\frac{a}{b^3+54}+\frac{b}{c^3+54}+\frac{c}{a^3+54}$
If $a+b+c = \dfrac{9}{2}$ and $a,b,c>0$, then what is the minimum value of $$\dfrac{a}{b^3+54}+\dfrac{b}{c^3+54}+\dfrac{c}{a^3+54} \qquad ?$$
My try:
$$\begin{align*}
\frac{a}{b^3+54}+\frac... | Point to note: The minimum value is not achieved when $a=b=c=\frac{3}{2}$, which gives a value of $\frac{4}{51}$. The minimum value is achieved at $a=\frac{3}{2}, b=3, c=0$, which gives $\frac{2}{27}$. Note that $\frac{4}{51}>\frac{2}{27}$. (and cyclic permutations)
We shall smooth towards this equality case.
Note: Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/357988",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
limit with $\arctan$ I have to find the limit
and want ask about a hint:
$$\lim_{n \to \infty} n^{\frac{3}{2}}[\arctan((n+1)^{\frac{1}{2}})- \arctan(n^{\frac{1}{2}})]$$
I dont have idea what to do. Derivatives and L'Hôpital's rule are so hard
| Write
$$\tan \left(\arctan\left( (n+1)^{1\over2}\right) - \arctan\left( n^{1\over2}\right) \right) = \frac{\sqrt{n+1}-\sqrt{n}}{1+\sqrt{n(n+1)}} = \frac{\sqrt{n}\left(\sqrt{1+{1\over n}}-1\right)}{1+\sqrt{n(n+1)}}$$
Thus
$$\arctan\left( (n+1)^{1\over2}\right) - \arctan\left( n^{1\over2}\right) = \arctan \left( \frac{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/358368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
Find this $\frac{1}{2m+1}+\frac{1}{2n+1}=\frac{2}{2k+1}$ Let $m,n,k\in \mathbb{N}$, and $m,n,k\ge 1,m\neq n\neq k\neq m $, such that
$$
\dfrac{1}{2m+1}+\dfrac{1}{2n+1}=\dfrac{2}{2k+1}
$$
Is there a solution? Or does this not have any solution?
| $\dfrac{1}{2m+1} +\dfrac{1}{2n+1}$, letting $2m+1$ to be the LCM of $2m+1$ and $2n+1$, Assume $2m+1=(2n+1) \cdot k$, your expression becomes $\dfrac{k+1}{2m+1}=\dfrac{2}{2l+1}$
$\dfrac{(k+1)}{2m+1}=\dfrac{2}{2l+1} \implies k+1=\dfrac{2(2m+1)}{2l+1}$. You just need a condition that $2l+1|2m+1$. Do you see that infinitel... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/358479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Residue of hyperbolic function How would I find the residue at $z_0=0$ of $$f(z)=\frac{\sinh(z)}{z^4(1-z^2)}$$I tried writing it as a series and reach $$\frac{1}{1-z^2}\sum_{n=0}^\infty \frac{z^{2n-3}}{(2n+1)!}$$ and then don't know where to go form there. Any help/hints appreciated.
| $$\sinh z = z + \frac{z^3}{3!} + \frac{z^5}{5!} + \cdots$$
$$\frac{1}{z^4}\sinh z = \frac{1}{z^3} + \frac{1}{z3!} + \frac{z}{5!} + \cdots$$
Now,
$$\frac{1}{1-z^2} = 1 + z^2 + z^4 + \cdots$$
So then, multiplying the series (note we only want to the the $z^{-1}$ terms, so no need to multiply every term)
$$\frac{1}{z^4(1-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/358566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
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.