Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
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Calculating $\lim_{x \rightarrow 1}(\frac{23}{1-x^{23}} - \frac{31}{1-x^{31}})$ How to calculate following limit?
$$\lim_{x \rightarrow 1}\left(\frac{23}{1-x^{23}} - \frac{31}{1-x^{31}}\right)$$
| Alternatively, applying L'Hopital's rule, we get:
\begin{align}
\lim_{x \to 1}{\frac{23}{1-x^{23}}-\frac{31}{1-x^{31}}}& =\lim_{x \to 1}{\frac{23(x^{23}+x^{24}+ \ldots +x^{30})-8(1+x+\ldots +x^{22})}{(1-x^{31})(1+x+ \ldots +x^{22})}} \\
& =\lim_{x \to 1}{\frac{23(23x^{22}+24x^{23}+ \ldots +30x^{29})-8(1+ 2x+\ldots +22x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/335423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
Prove $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ Let $a,b,c$ are non-negative numbers, such that $a+b+c = 3$.
Prove that $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$
Here's my idea:
$\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$
$2(\sqrt{a} + \sqrt{b} + \sqrt{c}) \ge 2(ab + bc + ca)$
$2(\sqrt{a} + \sqrt... | Since $ab+ac+bc=\frac{(a+b+c)^2-a^2-b^2-c^2}{2}=\frac{9-a^2-b^2-c^2}{2}$, we need to prove that
$$\sum_{cyc}\left(\sqrt{a}+\frac{a^2}{2}-\frac{3}{2}\right)\geq0$$ or
$$\sum_{cyc}(a^2+2\sqrt{a}-3)\geq0$$ or
$$\sum_{cyc}(\sqrt{a}-1)(a\sqrt{a}+a+\sqrt{a}+3)\geq0$$ or
$$\sum_{cyc}\left((\sqrt{a}-1)(a\sqrt{a}+a+\sqrt{a}+3)-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/336362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 4,
"answer_id": 1
} |
Surface integral over ellipsoid I've problem with this surface integral:
$$
\iint\limits_S {\sqrt{ \left(\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4}\right)}}{dS}
$$, where
$$
S = \{(x,y,z)\in\mathbb{R}^3: \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}= 1\}
$$
| What an interesting integral. I had to resort to referring to the first differential form of the spherical parametrization, but doing that, I am amazed at how this turns out.
We parametrize in the usual way:
$$x=a\sin{u} \cos{v}$$
$$y=b\sin{u} \sin{v}$$
$$z=c \cos{u}$$
where $u \in [0,\pi)$ and $v \in [0,2 \pi)$. The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/338155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 0
} |
If $2x = y^{\frac{1}{m}} + y^{\frac{-1}{m}}(x≥1) $ then prove that $(x^2-1)y^{"}+xy^{'} = m^{2}y$ How do I prove following?
If $2x = y^{\frac{1}{m}} + y^{\frac{-1}{m}},(x≥1)$, then prove that $(x^2-1)y^{"}+xy^{'} = m^{2}y$
| Complete the square:
$2x = y^\frac{1}{m} + y^\frac{-1}{m}$
$2xy^\frac{1}{m} = y^\frac{2}{m} + 1$
$0 = y^\frac{2}{m} - 2xy^\frac{1}{m} + 1$
$0 = (y^\frac{2}{m} - 2xy^\frac{1}{m} + x^2) - x^2 + 1$
$0 = (y^\frac{1}{m} - x)^2 - x^2 + 1$
$(y^\frac{1}{m} - x)^2 = x^2 - 1$
$y^\frac{1}{m} - x = \sqrt{x^2 - 1}$
$y^\frac{1}{m} =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/338492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Find a closed form of the series $\sum_{n=0}^{\infty} n^2x^n$ The question I've been given is this:
Using both sides of this equation:
$$\frac{1}{1-x} = \sum_{n=0}^{\infty}x^n$$
Find an expression for $$\sum_{n=0}^{\infty} n^2x^n$$
Then use that to find an expression for
$$\sum_{n=0}^{\infty}\frac{n^2}{2^n}$$
This is a... | Hint: $n^2=n(n-1)+n$ and $x^2 x^{n-2}=x^n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/338852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 3
} |
Solving linear recurrence relation Solve the following linear recurrence relation:
$$h_n=4h_{n-1}-4h_{n-2}+n^2 2^n$$
for $n\geq2$ and $h_0=h_1=1$
| Use generating functions. Define $H(z) = \sum_{n \ge 0} h_n z^n$, and write the recurrence as:
$$
h_{n + 2} = 4 h_{n + 1} - 4 h_n + 4 (n + 2)^2 2^n
$$
By the properties of ordinary generating functions (see e.g. Wilf's "generatingfunctionology") with the operator $z D = z \dfrac{d}{d z}$
:
$$
\begin{align*}
\frac{H(z) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/339975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
How to solve $5^{2x}-3\cdot2^{2y}+5^x\cdot2^{y-1}-2^{y-1}-2\cdot5^x+1=0$ in $\mathbb{Z}$ how to solve in $\Bbb Z$:
$$5^{2x}-3\cdot2^{2y}+5^x\cdot2^{y-1}-2^{y-1}-2\cdot5^x+1=0$$
| Note that $5^{2x}-2\cdot5^x+1=(5^x-1)^2$, and $5^x\cdot2^{y-1}-3\cdot2^{2y}-2^{y-1}=(5^x-1)2^{y-1}-3\cdot2^{2y}$, so the original equation can be written
$$(5^x-1)(5^x-1+2^{y-1})-3\cdot2^{2y}=0\;.\tag{1}$$
The recurring element $5^x-1$ suggests making a substitution $u=5^x-1$, and since $3\cdot2^{2y}$ can be expressed ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/341536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove that $\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$ $$\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$$
I have attempted this question by expanding the left side using the cosine sum and difference formulas and then multiplying, and then simplifying till I replicated the identity on the right. I am not s... | For a slightly different solution,
$2\cos^2 A-2\sin^2 B$
$=2\cos^2 A-1+1-2\sin^2 B$
$=\cos 2A+\cos 2B$
$=2\cos (A+B) \cos (A-B)$
and halve each side.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/345703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 4
} |
Integrating $\int_0^\infty \sin(1/x^2) \, \operatorname{d}\!x$ How would one compute the following improper integral:
$$\int_0^\infty \sin\left(\frac{1}{x^2}\right) \, \operatorname{d}\!x$$
without any knowledge of Fresnel equations?
I was thinking of using the comparison theorem since $\sin x$ is never greater than ... | I prefer to evaluate the more general integral below
$$\displaystyle I(a):=\int_{-\infty}^{\infty} \sin \left(\frac{a}{x^2} \right)dx, \tag*{}$$
where $a\geq 0.$
Letting $y=\frac{1}{x^2}$ transforms the integral into
$\displaystyle \begin{aligned} I(a)&=2 \int_{\infty}^0 \sin (a y)\left(-\frac{1}{2 y^{\frac{3}{2}}} d ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/349612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
If $a,b$, and $c$ are reals satisfying $ \frac{b+c}{a}+\frac{a+c}{b}+\frac{a+b}{c}=6$, calculate $ x = \frac{(a+b+c)^3}{a^3+b^3+abc}$ $$ \frac{b+c}{a}+\frac{a+c}{b}+\frac{a+b}{c}=6$$
$$ x = \frac{(a+b+c)^3}{a^3+b^3+abc}$$
As a trivial solution, I found $$a=b=c$$
then, $x$ will always be $9$.
Despite this, my algebra te... | You can also have $a=b=c=-2$. In that case, $x$ is still $9$. Given the wording of the problem, the value of $x$ should not depend upon what solution you find for $a,b,c$, so you should be able to find one, evaluate $x$ from it, and quit. If they want yo to prove it independent of $a,b,c$ they should ask.
Added: $a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/350560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Stuck finding a recursive recurrence relation. I am analyzing the following algorithm:
QUANT(n):
if n == 0 or n == 1:
return 1
else
return (n-1)*QUANT(n-1) + n
I need to find the recurrence relation of this algorithm and prove it using mathematical induction. Here is what I have tried so far:
$$a_0 = 1, a_1 = 1,... | You have the recurrence:
$$
a_{n + 1} = n a_n + n + 1 \quad a_0 = 1
$$
This is a non-homogeneous linear first order recurrence. If you have:
$$
x_{n + 1} - u_n x_n = f_n
$$
Dividing by $u_n \ldots u_0$ gives:
$$
\frac{x_{n + 1}}{u_n \ldots u_0} - \frac{x_n}{u_{n - 1} \ldots u_0}
= \frac{f_n}{u_n \ldots u_0}
$$
This... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/350632",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
The question is whether the two values of $I_2$ as obtained below are same? Integrate, $\int \frac{\sin^{2}{2\theta}}{2\cos{2\theta}}d\theta$
Let, $\sin2\theta=t$
then, ${2\cos2\theta} d\theta=dt$
$dt=\frac{1}{2}\frac{1}{\sqrt{1-t^2}}$
Integral =$\frac{1}{4}\int {\frac{t^2}{1-t^2}} dt$
Integral =$\frac{1}{4}\int {\frac... | A few things to point out:
*
*$\int\frac{1}{x}dx=\ln|x|+C$.
*$\ln\frac{a}{b}=-\ln\frac{b}{a}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/350678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Determining series formula Is there any procedure to follow when determining the function of a series? This seems simple but for I can't figure it out.
$$ \frac15 + \frac18 + \frac1{11} +\frac1{14} + \frac1{17}+\cdots$$
| Generally speaking, when I see a series like this, I try to construct a generating function of some sort. In this case:
$$f(x) = \sum_{k=1}^{\infty} \frac{x^{3 k+2}}{3 k+2}$$
You can easily show that
$$f'(x) = \frac{x^4}{1-x^3}$$
So we integrate and use $f(0)=0$ to evaluate $f(1)$. In this case, it should be clear th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/354236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
How do I generate my linear transformation matrix in this example? Let $T$ be the linear transformation of space of polynomials $P^3$ given by
$$T(a + bx + cx^2) = a + b(x + 1) + c(x + 1)^2$$
Find all eigenvalues and eigenvectors of $T$.
| Hints: $a+bx+cx^2$ is equivalent to the vector $[a,b,c]^{T}$. Expanding the RHS
\begin{align}
T=(a+b+c)+(b+2c)x+cx^2
\end{align}
In terms of matrices, $T$ should be equivalent to
\begin{align}
\begin{bmatrix}
a+b+c \\ b+2c \\ c
\end{bmatrix}=
\begin{bmatrix}
1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1
\end{bmatrix}
\begin{bmat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/356726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding the maximum of $\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}$ If $a,b,c,d$ are distinct real numbers such that $\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}+\dfrac{d}{a}=4$ and $ac=bd$.
Then how would we calculate the maximum value of $$\dfrac{a}{c}+\dfrac{b}{d}+\dfrac{c}{a}+\dfrac{d}{b}.$$
I was unable to proceed... | Substituting $d=\frac{ac}{b}$ the constraint becomes
$$\left(\frac{a}{b}+\frac{b}{a}\right)+\left(\frac{c}{b}+\frac{b}{c}\right)=4$$
while the function to be maximized becomes
$$\frac{a}{c}+\frac{b^2}{ac}+\frac{c}{a}+\frac{ac}{b^2}=\frac{b}{c}\left(\frac{a}{b}+\frac{b}{a}\right)+\frac{c}{b}\left(\frac{a}{b}+\frac{b}{a}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/358223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Laurent series $z=i$ To find the Laurent series expansion for $\frac{1}{1+z^2}$, centered at $z=i$ would using partial fraction decomposition be the right idea? So, $\frac{1}{1+z^2}$=$\frac{1}{(z+i)(z-i)}$=$\frac{\frac{-1}{2i}}{z+i} +\frac{\frac{1}{2i}}{z-i}$?
| First:
$$\frac{1}{z^2+1}=\frac{1}{(z-i)(z+i)}=\frac{1}{2i}\left(\frac{1}{z-i}-\frac{1}{z+i}\right)$$
Now
$$|z-i|<1\implies\frac{1}{z+i}=\frac{1}{z-i+2i}=\frac{1}{2i}\frac{1}{1+\frac{z-i}{2i}}=\frac{1}{2i}\left(1-\frac{z-i}{2i}+\frac{(z-i)^2}{-4\cdot 2!}+\ldots\right)\implies$$
$$\frac{1}{z^2+1}=\frac{1}{2i}\left(\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/358840",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
The right way to calculate the volume obtained by rotating the area between 2 graphs around the x axis If i have 2 graphs: $f(x)=x\cdot \frac{\sqrt{1-x^{2}}}{2},\:g(x)=\frac{\sqrt{1-x^{2}}}{2}$
And need to calculate the volume obtained by rotating the area between $f(x)$ and $g(x)$
Around the $x$-axis.
I know i need t... | Since you have concerns, we do the calculation. We have $g^2(x)=\frac{1-x^2}{4}$ and $f^2(x)=\frac{x^2(1-x^2)}{4}$, and therefore the difference is $\frac{(1-x^2)^2}{4}$, which is $\frac{1}{4}(1-2x^2+x^4)$. Integrate from $0$ to $1$. An antiderivative is $\frac{1}{4}(x-\frac{2}{3}x^3+\frac{1}{5}x^5)$.
Do the substitut... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/359682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove that $\tan(75^\circ) = 2 + \sqrt{3}$ My (very simple) question to a friend was how do I prove the following using basic trig principles:
$\tan75^\circ = 2 + \sqrt{3}$
He gave this proof (via a text message!)
$1. \tan75^\circ$
$2. = \tan(60^\circ + (30/2)^\circ)$
$3. = (\tan60^\circ + \tan(30/2)^\circ) / (1 - \ta... | The formula you want to see is: $\tan(x+y)=\frac{\tan(x)+\tan(y)}{1-\tan(x)\tan(y)}$ for any degrees $x$ and $y$.
On the other hand, proving this tangent equality from the formulas $\sin(x+y)=\sin(x)\cos(y)+\sin(y)\cos(x)$ and $\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)$ will be a good exercise for a beginner.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/360747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 1
} |
A sufficient condition to ensure $\alpha=\beta$ Let $\alpha$, $\beta$ be acute angles satisfying
$$
\frac{\sin 2\alpha}{\sin(2\alpha+\beta)}=\frac{\sin2\beta}{\sin(2\beta+\alpha)}
$$
Show then $\alpha=\beta$.
| I got a solution for this problem. Here we have to use the following identities
\begin{eqnarray*}
\sin A\sin B&=&-\frac{1}{2}[\cos(A+B)-\cos(A-B)],\\
\cos A-\cos B&=&-2\sin\frac{A+B}{2}\sin\frac{A-B}{2},\\
\sin 3A&=&3\sin A-4\sin^3 A,\\
\sin 5A&=&16\sin^5 A-20\sin^3 A+5\sin A.
\end{eqnarray*}
We obtain
\begin{eqnarray*... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/363331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Minimal Distance between two curves What is the minimal distance between curves?
*
*$y = |x| + 1$
*$y = \arctan(2x)$
I need to set a point with $\cos(t), \sin(t)$?
| Let $(a,|a|+1)$ be a point on the first curve and let $(b,\arctan(2b))$ be a point on the second curve.
Half the distance between the two points squared is $$\frac{1}{2}d^2 = \frac{1}{2}(a-b)^2+\frac{1}{2}(|a|+1-\arctan(2b))^2.$$
To find the minimum of this expression we set the partial derivatives to zero:
$$\frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/364341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Pythagorean triples and perfect squares This problem is giving me difficulty:
Show that in any Pythagorean triple there exist at most a single perfect square
So far I've been working with the equations for primitive Pythagorean Triples (ie $x = m^2 - n^2$, $y = 2mn$, $z = m^2 + n^2$) but that hasn't really worked out... | Are you familiar with the following?
The area of a Pythagorean triangle is never square.
For a proof, let $H$ be the set of $h>0$ such that there exists a
primitive Pythagorean triangle of hypotenuse $h$ whose area is square. Suppose that
$H\ne\emptyset$, then we can take $h_0=\min\left(H\right)$, and write
$h_0^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/365445",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
$f: \mathbb{R} \to \mathbb{R}$ satisfies $(x-2)f(x)-(x+1)f(x-1) = 3$. Evaluate $f(2013)$, given that $f(2)=5$ The function $f : \mathbb{R} \to \mathbb{R}$ satisfies $(x-2)f(x)-(x+1)f(x-1) = 3$. Evaluate $f(2013)$, given that $f(2) = 5$.
| Let $x-1=y$, then,
$$(y-1)f(y+1)-(y+2)f(y)=3$$
$$\implies f(y+1)=\frac{3+(y+2)f(y)}{y-1} \;\;\;\;\;(1)$$
Lemma: $\forall \; n \geq 2 \in \mathbb{N}$, $f(n)=n(n-1)(n+1)-1$.
Base Case: If $n=2$ then $f(n)=1 \cdot 2 \cdot 3 -1=5$ which is true by information provided in the question.
Inductive Step: Assume for $n=k$ that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/367473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
} |
Evaluating a trigonometric integral using residues Finding the trigonometric integral using the method for residues:
$$\int_0^{2\pi} \frac{d\theta}{ a^2\sin^2 \theta + b^2\cos^2 \theta} = \frac{2\pi}{ab}$$ where $a, b > 0$.
I can't seem to factor this question
I got up to $4/i (z) / ((b^2)(z^2 + 1)^2 - a^2(z^2 - 1)^2 ... | Without residues and same dirty trigonometric trick as before:
$$\frac{1}{a^2\sin^2t+b^2\cos^2t}=\frac{1}{b^2\cos^2t}\frac{1}{1+\frac{a^2}{b^2}\tan^2t}=\frac{1}{ab}\frac{\frac{a}{b\cos^2t}}{1+\left(\frac{a}{b}\tan t\right)^2}$$
and since
$$\frac{a}{b\cos^2t}=\left(\frac{a}{b}\tan t\right)'$$
we finally get:
$$\int\lim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/367798",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Symmetry properties of $\sin$ and $\cos$. Why does $\cos\left(\frac{3\pi}{2} - x\right) = \cos\left(-\frac{\pi}{2} - x\right)$? For a question such as:
If $\sin(x) = 0.34$, find the value of $\cos\left(\frac{3\pi}{2} - x\right)$.
The solution says that:
\begin{align*}
\cos\left(\frac{3\pi}{2} - x\right) &= \cos\left... | For your first question, recall that $-\frac{\pi}{2}$ is just a different "angle name" for the same direction as $\frac{3\pi}{2}$. The first is obtained by turning clockwise from the positive x-axis ($\theta = 0$) by $90^{\circ}$ or $\frac{\pi}{2}$ radians, and the second by turning counter-clockwise from $\theta = 0$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/367938",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
What is going on with this constrained optimization? I'd like to figure out what is going on when trying to maximize a function (below $a_i$ are real numbers)
$F = a_1a_2 + a_2a_3 + \cdots + a_{n - 1}a_n + a_na_1;$
When we have active constraints
$h_1 = a_1 + a_2 + \cdots + a_n = 0;$
$h_2 = a_1^2 + a_2^2 + \cdots +a_n... | Well, on adding up the n equations you have, you get
$$2\sum_{i=1}^{n}a_i = n\lambda_1 + 2\lambda_2\sum_{i=1}^{n}a_i$$
which gives, $\lambda_1 = 0$ as $\sum_{i=1}^{n}a_i=0$.
So now you need to find the nullspace of the matrix, which should hopefully be easier?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/370251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the polar form of $ z = 1- \sin (\alpha) + i \cos (\alpha) $? How do I change $ z = 1- \sin (\alpha) + i \cos (\alpha) $ to polar? I got $r = (2(1-\sin(\alpha))^{\frac{1}{2}} $. I have problems with the exponential part. What should I do now?
| $1-\sin\alpha\ge 0$
If $1-\sin\alpha=0,\cos\alpha=0, z=0+i\cdot0$
If $1-\sin\alpha> 0$
Let $1-\sin\alpha=r\cos t, \cos\alpha=r\sin t$ where $r> 0$
Squaring & adding we get
$r^2=2(1-\sin\alpha)=2(\cos\frac\alpha2-\sin\frac\alpha2)^2=4\cos^2(\frac\pi4+\frac\alpha2)$
On division, $\frac{r\sin t}{r\cos t}=\frac{\cos \alp... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Does one always use augmented matrices to solve systems of linear equations? The homework tag is to express that I am a student with no working knowledge of math.
I know how to use elimination to solve systems of linear equations. I set up the matrix, perform row operations until I can get the resulting matrix into row... | Augmenting the matrix is just notational shorthand.
When you do row operations, what you're really doing is multiplying both sides of an equation by some matrix $P_1$:
$$\begin{align*}
Ax &= b \\
P_1Ax &= P_1b \tag{$P_1$ represents some row operation} \\
P_2P_1Ax &= P_2 P_1 b \tag{$P_2$ represents some row operation} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/373667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Finding the remainder when $2^{100}+3^{100}+4^{100}+5^{100}$ is divided by $7$
Find the remainder when $2^{100}+3^{100}+4^{100}+5^{100}$ is divided by $7$.
Please brief about the concept behind this to solve such problems. Thanks.
| The idea is to reach at $\equiv\pm1\pmod n$ for a given modulo integer $n$
In general we should utilize Fermat's little theorem for prime modulo
or Euler's Totient Theorem or Carmichael Function for non-prime modulo
unless we can reach $\pm1$ easily like below.
$2^3=8\equiv1\pmod 7\implies 2^{100}=2\cdot (2^3)^{33}\eq... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
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Discrete Time Fourier Transform example: $x = [1 \; 2 \; 3 \; 4]^T \; \rightarrow \; X=?$ How do I find the Discrete Fourier Transform of the sequence below?
$$ x = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix}$$
Show all steps.
| Apply Discrete Time Fourier Transformation on the vector
$$ x = \begin{bmatrix} x[0] \\ x[1] \\ x[2] \\ x[3] \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix} $$
to find the frequency domain vector
$$ \mathcal{F} \left \{ x \right \} = X = \begin{bmatrix} X[0] \\ X[1] \\ X[2] \\ X[3] \end{bmatrix}. $$
Use ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Complex roots of polynomial equations with real coefficients Consider the polynomial $x^5 +ax^4 +bx^3 +cx^2 +dx+4$ where $a, b, c, d$ are real
numbers. If $(1 + 2i)$ and $(3 - 2i)$ are two roots of this polynomial then what is the value of $a$ ?
| As the coefficients of the different powers of $x$ are real, all the complex roots must occur with conjugate pair.
So, the other two roots are $1-2i,3+2i$
So if $p$ is the fifth root,
using Vieta's Formulas $(1-2i)(3+2i)(1+2i)(3-2i)p=(-1)^5\frac41$
and $(1-2i)+(3+2i)+(1+2i)+(3-2i)+p=-\frac a1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/381114",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Prove by mathematical induction that $1 + 1/4 +\ldots + 1/4^n \to 4/3$ Please help. I haven't found any text on how to prove by induction this sort of problem:
$$
\lim_{n\to +\infty}1 + \frac{1}{4} + \frac{1}{4^2} + \cdots+ \frac{1}{4^n} = \frac{4}{3}
$$
I can't quite get how one can prove such. I can prove basic di... | I'm not really sure if it is possible to prove this using induction but we can do something else.
The sum $\displaystyle 1 + \frac{1}{4} + \frac{1}{4^2} + \cdots \frac{1}{4^k}$ can be written as $\displaystyle\sum_{n=0}^k \frac{1}{4^n}$. Now $\displaystyle\sum_{n=0}^k \displaystyle\frac{1}{4^n} = \frac{1-\frac{1}{4^{k+... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 4
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How to write this conic equation in standard form? $$x^2+y^2-16x-20y+100=0$$
Standard form? Circle or ellipse?
| $$
\begin{align}
x^2+y^2-16x-20y+100 & = 0 \\ \\
\color{blue}{\bf x^2 -16x} + \color{red}{\bf y^2 -20y } + 100 & = 0
\end{align}
$$
We complete the square: $$x^2+bx+(b/2)^2-(b/2)^2+c= (x+b/2)^2+c-(b/2)^2$$
$$\begin{align}
\color{blue}{\bf x^2 - 16x} + \underbrace{\bf \left(\frac{-16}{2}\right)^2}_{\color{green}{\bf \l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/382554",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Integrating a school homework question. Show that $$\int_0^1\frac{4x-5}{\sqrt{3+2x-x^2}}dx = \frac{a\sqrt{3}+b-\pi}{6},$$ where $a$ and $b$ are constants to be found.
Answer is: $$\frac{24\sqrt3-48-\pi}{6}$$
Thank you in advance!
| $$I=\int_0^1\frac{4x-5}{\sqrt{3+2x-x^2}}dx = \int_0^1\frac{4x-5}{\sqrt{4-(x-1)^2}}dx$$
Let $x-1=2\sin\theta\implies dx=2\cos\theta d\theta$
If $x=0, \sin\theta=\frac12, \theta=\frac\pi6$
If $x=1, \sin\theta=0, \theta=0$
$$\text{So,}I=\int_{\frac\pi6}^0\frac{4(2\sin\theta+1)-5}{2\cos\theta}\cdot2\cos\theta d\theta$$
$$=... | {
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"url": "https://math.stackexchange.com/questions/384530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$\frac{1}{4^n}\binom{1/2}{n} \stackrel{?}{=} \frac{1}{1+2n}\binom{n+1/2}{2n}$ - An identity for fractional binomial coefficients In trying to write an answer to this question:
calculate the roots of $z = 1 + z^{1/2}$ using Lagrange expansion
I have come across the identity
$$
\frac{1}{4^n}\binom{1/2}{n} = \frac{1}{1+... | Simply compute ratios of each side. Introducing notation $\Theta_n f(n) = \frac{f(n+1)}{f(n)}$:
$$
\Theta_n \frac{1}{4^n} \binom{1/2}{n} = \frac{1}{4} \Theta_n \binom{1/2}{n} =\frac{1}{4} \frac{\Gamma(n+1) \Gamma\left(\frac{3}{2}-n\right)}{\Gamma(n+2) \Gamma\left(\frac{1}{2}-n\right)} = -\frac{2n-1}{8(n+1)}
$$
On a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/384731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Find the following integral: $\int {{{1 + \sin x} \over {\cos x}}dx} $ My attempt:
$\int {{{1 + \sin x} \over {\cos x}}dx} $,
given : $u = \sin x$
I use the general rule:
$\eqalign{
& \int {f(x)dx = \int {f\left[ {g(u)} \right]{{dx} \over {du}}du} } \cr
& {{du} \over {dx}} = \cos x \cr
& {{dx} \over {du}} = ... | Do this:
$ \int { \frac { 1+\sin { x } }{ \cos { x } } dx } =\int { \frac { \left( 1+\sin { x } \right) \left( 1-\sin { x } \right) }{ \cos { x } \left( 1-\sin { x } \right) } dx= } \int { \frac { \cos { x } }{ \left( 1-\sin { x } \right) } } dx$
Then do the substitution rule
$\left( 1-\sin { x } \right)... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 0
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Finding the sum of the coefficients of polynomial of degree 21 Problem : Find the sum of the coefficients of the polynomial$ p(x) =(3x-2)^{17}(x+1)^4$
Solution :
$ p(x) =(3x-2)^{17}(x+1)^4 $
$= (a_0+a_1x+....a_{17}x^{17})(b_0+b_1x+....b_4x^4)$ for some $a_i; b_j$
= $(c_0+c_1x+...c_{21}x^{21})$ for some $c_k$
Sum of t... | If $P(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + \cdots + a_{0}$, and we want to find $a_{n} + a_{n-1} + \cdots + a_{1} + a_{0}$, then we see that:
$P(1) = a_{n}(1)^{n} + a_{n-1}(1)^{n-1} + \cdots + a_{1}(1) + a_{0} = a_{n}(1) + a_{n-1}(1) + \cdots + a_{1}(1) + a_{0} = a_{n} + a_{n-1} + \cdots + a_{1} + a_{0}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/387222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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How to simplify $\frac{(\sec\theta -\tan\theta)^2+1}{\sec\theta \csc\theta -\tan\theta \csc \theta} $ How to simplify the following expression :
$$\frac{(\sec\theta -\tan\theta)^2+1}{\sec\theta \csc\theta -\tan\theta \csc \theta} $$
| Write $1$ in the numerator as : $$\sec^2(\theta) - \tan^2(\theta)$$
$$\frac{(\sec\theta -\tan\theta)^2+\sec^2\theta - \tan^2\theta}{\sec\theta \csc\theta -\tan\theta \csc \theta} $$
$$\frac{(\sec\theta -\tan\theta)^2+(\sec\theta - \tan\theta)(\sec\theta + \tan\theta)}{\sec\theta \csc\theta -\tan\theta \csc \theta} $$
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/387427",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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What is the centroid of a hollow spherical cap? I have a unit hollow sphere which I cut along a diameter to generate two equivalent hollow hemispheres. I place one of these hemispheres on an (x,y) plane, letting it rest on the circular planar face where the cut occurred.
If the hemisphere was solid, we could write th... | In Cartesian coordinates, the hemisphere can be represented by a rotation about the z-axis of a circle, for which we can use, say, $y^2 + z^2 = 1$ to produce the infinitesimal "belt" of surface area $ dS = 2 \pi y \ ds \ $ from
$$\frac{d}{dz} [y^2 + z^2] \ = \ \frac{d}{dz} [1] \ \Rightarrow \ \frac{dy}{dz} \ = \ -\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/388083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Closed form for $\sum_{n=1}^\infty\frac{1}{2^n\left(1+\sqrt[2^n]{2}\right)}$ Here is another infinite sum I need you help with:
$$\sum_{n=1}^\infty\frac{1}{2^n\left(1+\sqrt[2^n]{2}\right)}.$$
I was told it could be represented in terms of elementary functions and integers.
| Note that
$$\frac{2^{-n}}{2^{2^{-n}}-1}-\frac{2^{-(n-1)}}{2^{2^{-(n-1)}}-1} = \frac{2^{-n}}{2^{2^{-n}}+1} $$
Thus we have a telescoping sum. However, note that
$$\lim_{n \to \infty} \frac{2^{-n}}{2^{2^{-n}}-1} = \frac{1}{\log{2}}$$
Therefore the sum is
$$a_1-a_0 + a_2-a_1 + a_3-a_2 + \ldots + \frac{1}{\log{2}} = \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/390801",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "34",
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Contour Integral and Complex Identity While studying I came across this problem:
(a) For $z=x+iy$, show that
$$|\cos \pi z|^2=\frac{1}{2}(\cos(2\pi x)+\cosh(2\pi y))$$
(b) For a positive integer let $\gamma N$ be the square connecting the points $\pm N\pm Ni$ oriented in positive direction. Show that
$$|\cos \pi z|\ge... | For (a), use the cosine addition formula: $\cos{(a+i b)} = \cos{a} \cos{i b} - \sin{a} \sin{i b}$. Note that $\cos{i b} = (e^{i (i b)} + e^{-i (i b)})/2 = (e^{-b}+e^b)/2 = \cosh{b}$. Similarly, you can show that $\sin{i b} = i \sinh{b}$. Then
$$\cos{(a+i b)} = \cos{a} \cosh{b} - i \sin{a} \sinh{b}$$
so that
$$\begin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/390925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Reducibility of $x^{2n} + x^{2n-2} + \cdots + x^{2} + 1$ Just for fun I am experimenting with irreducibility of certain polynomials over the integers. Since $x^4+x^2+1=(x^2-x+1)(x^2+x+1)$, I thought perhaps $x^6+x^4+x^2+1$ is also reducible. Indeed:
$$x^6+x^4+x^2+1=(x^2+1)(x^4+1)$$
Let $f_n(x)=x^{2n}+x^{2n-2}+\cdots + ... | For odd $n$ you can always write:
$$f_n(x) = \sum_{i=0}^n x^{2i} = (x^2+1)(x^{2n-2}+x^{2n-6}+x^{2n-10}+...)$$
For even $n$ this trick no longer works.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Solve $\sqrt{2x-5} - \sqrt{x-1} = 1$ Although this is a simple question I for the life of me can not figure out why one would get a 2 in front of the second square root when expanding. Can someone please explain that to me?
Example: solve $\sqrt{(2x-5)} - \sqrt{(x-1)} = 1$
Isolate one of the square roots: $\sqrt{(2x-5... | It is an identity:
$$(1+x)^2=1+2x+x^2$$
How? Well, consider this:
$$\begin{equation*}
\begin{split}
(1+x)^2&=(\color{blue}{1}+\color{red}{x})(1+x)\\
&=\color{blue}{1}(1+x)+\color{red}{x}(1+x)\\
&=1+x+x+x^2\\
&=1+2x+x^2\\
\end{split}
\end{equation*}$$
In general, $(a+b)^2=a^2+2ab+b^2$, which you can try to deduce yourse... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 4
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How to integrate $\int \sqrt{x^2+a^2}dx$ $a$ is a parameter. I have no idea where to start
| Since $\sinh^2(x)+1
=\cosh^2(x)
$,
this suggests
letting $x = a \sinh(y)$.
$dx = a \cosh(y)$
and $x^2+a^2 = a^2(\sinh^2(y)+1)
= a^2 \cosh^2(y)$,
so $a \cosh(y) = \sqrt{x^2+a^2}$.
$\begin{align}
\int{\sqrt{x^2+a^2}}dx
&=\int a^2 \cosh^2(y) dy\\
&= a^2 \int (e^{2y}+2+e^{-2y})\,dy/4\\
&= (a^2/4) (e^{2y}/2 + 2y - e^{-2y}/2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/392663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
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positive Integer value of $n$ for which $2005$ divides $n^2+n+1$
How Can I calculate positive Integer value of $n$ for which $2005$
divides $n^2+n+1$
My try:: $2005 = 5 \times 401$
means $n^2+n+1$ must be a multiple of $5$ or multiple of $401$
because $2005 = 5 \times 401$
now $n^2+n+1 = n(n+1)+1$
now $n(n+1)+1$ co... | If a number is divisible by $2005=5\cdot401,$ it must be divisible by $5$ and by $401$
Now, as you have identified $n^2+n+1$ is not divisible by $5,$ it can not be divisible by any multiple of $5$ like $2005=5\cdot401$
Alternatively,
$$n^2+n+1\equiv0\pmod {2005}\implies n^2+n+1\equiv0\pmod 5$$
$$\implies 4(n^2+n+1)\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/393299",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Comparing $\sqrt{1001}+\sqrt{999}\ , \ 2\sqrt{1000}$ Without the use of a calculator, how can we tell which of these are larger (higher in numerical value)?
$$\sqrt{1001}+\sqrt{999}\ , \ 2\sqrt{1000}$$
Using the calculator I can see that the first one is 63.2455453 and the second one is 63.2455532, but can we tell with... | The answer is: YES, we can!
$$
\begin{align}
(\sqrt{1001}+\sqrt{999})^2&=2000+2\sqrt{1001\times 999}
\\
&=2000+2\sqrt{(1000+1)(1000-1)}
\\
&=2000+2\sqrt{1000^2-1}
\end{align}
$$
\begin{align}
\text{and that: }(2\sqrt{1000})^2&=4000
\\
&=2000+2\sqrt{1000^2}
\end{align}
Since $2000+2\sqrt{1000^2-1}<2000+2\sqrt{1000^2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/394648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 11,
"answer_id": 8
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Evaluating $\int_0^\infty \frac{\log (1+x)}{1+x^2}dx$ Can this integral be solved with contour integral or by some application of residue theorem?
$$\int_0^\infty \frac{\log (1+x)}{1+x^2}dx = \frac{\pi}{4}\log 2 + \text{Catalan constant}$$
It has two poles at $\pm i$ and branch point of $-1$ while the integral is to b... | Setting $a=1$ in
$$ \int_0^\infty\frac{\operatorname{Li}_a(-x)}{1+x^2}\mathrm{d}x=-2^{-a-1}\pi\, \eta(a)-a\beta(a+1)$$
we have
$$\int_0^\infty\frac{\operatorname{Li}_1(-x)}{1+x^2}\mathrm{d}x=-\int_0^\infty\frac{\ln(1+x)}{1+x^2}\mathrm{d}x=-\frac14\ln(2)-\beta(2)=-\frac14\ln(2)-G$$
Other results:
\begin{gather*}
\int_... | {
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"url": "https://math.stackexchange.com/questions/396170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 9,
"answer_id": 6
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A special curve $\left( \frac { x^{ 2 }+1 }{ 1-x^{ 2 } } \right)$. Let $a,x\in \mathbb{Q}$ and $n\in\mathbb{N}$
For $0<x<1$, $a>0$ and $n>1$, show that
$$\ \frac { x^{ 2 }+1 }{ 1-x^{ 2 } } \neq { a }^{ n }$$.
For example, for $x = 0.3$, then
$\frac {109 }{ 91 }$ can not be shown as $a^2$ or $a^3$ etc.
| Assume on the contrary that $\dfrac{1+x^2}{1-x^2}=a^n$ for some $x, a \in \mathbb{Q}, n \in \mathbb{N}, 0<x<1, a>0, n>1$. Write $x=\frac{s}{t}, s<t, s, t \in \mathbb{Z}^+, \gcd(s, t)=1$, so $\dfrac{t^2+s^2}{t^2-s^2}=a^n$
We consider 2 cases:
Case 1: $s, t$ have different parities.
Then $\gcd(t^2+s^2, t^2-s^2)=1$, so w... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Non-linear first order differential equation I've found this particular equation rather tough, can you give me some hints on how to solve
$$\dot{y}+t\cos\frac{\pi y}{2}+1-t=y$$
Thanks a lot.
| $\dot{y}+t\cos\dfrac{\pi y}{2}+1-t=y$
$\dfrac{dy}{dt}=\left(1-\cos\dfrac{\pi y}{2}\right)t+y-1$
Let $u=y-1$ ,
Then $y=u+1$
$\dfrac{dy}{dt}=\dfrac{du}{dt}$
$\therefore\dfrac{du}{dt}=\biggl(1-\cos\dfrac{\pi(u+1)}{2}\biggr)t+u$
$\left(\left(1-\cos\left(\dfrac{\pi u}{2}+\dfrac{\pi}{2}\right)\right)t+u\right)\dfrac{dt}{du}=... | {
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"url": "https://math.stackexchange.com/questions/398818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
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Solve the equation $\sqrt{3x-2} +2-x=0$
Solve the equation: $$\sqrt{3x-2} +2-x=0$$
I squared both equations $$(\sqrt{3x-2})^2 (+2-x)^2= 0$$
I got $$3x-2 + 4 -4x + x^2$$
I then combined like terms $x^2 -1x +2$
However, that can not be right since I get a negative radicand when I use the quadratic equation.
$x = 1/2... | What you did was not squaring both sides of the equation. If we have an equation
$$a+b=0,$$
then squaring both sides produces
$$\begin{align*}
(a+b)^2&=0^2\\a^2+2ab+b^2&=0.
\end{align*}$$
In your case, you have $a=\sqrt{3x-2}$ and $b=2-x$. But when you squared the left side of the equation, what you've written is inste... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/398984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to solve this system of equation. $x^2-yz=a^2$
$y^2-zx=b^2$
$z^2-xy=c^2$
How to solve this equation for $x,y,z$. Use elementary methods to solve (elimination, substitution etc.).
Given answer is:$x=\pm\dfrac{a^4-b^2c^2}{\sqrt {a^6+b^6+c^6-3a^2b^2c^2}}\,$, $y=\pm\dfrac{b^4-a^2c^2}{\sqrt {a^6+b^6+c^6-3a^2b^2c^2}}\,$ ... | Subtract them we get
$$(x^2-y^2)+xz-yz = a^2-b^2 \implies (x-y)(x+y+z) = (a-b)(a+b)$$
$$(y^2-z^2)+yx-zx = b^2-c^2 \implies (y-z)(x+y+z) = (b-c)(b+c)$$
$$(z^2-x^2)+zy-xy = c^2-a^2 \implies (z-x)(x+y+z) = (c-a)(c+a)$$
Now consider the cases $x+y+z=0$ and $x+y+z\neq 0$. I trust you can finish it off from here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/401436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Listing subgroups of a group I made a program to list all the subgroups of any group and I came up with satisfactory result for $\operatorname{Symmetric Group}[3]$ as
$\left\{\{\text{Cycles}[\{\}]\},\left\{\text{Cycles}[\{\}],\text{Cycles}\left[\left(
\begin{array}{cc}
1 & 2 \\
\end{array}
\right)\right]\right\},\left... | By using GAP, you can make them all so easier and visible.
gap> LoadPackage("sonata");;
gap> s4:=SymmetricGroup(IsPermGroup,4);;
gap> g:=Subgroups(s4);;
gap> List([1..Size(g)],k->g[k]);
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/401971",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Show that $n \ge \sqrt{n+1}+\sqrt{n}$ (how) Can I show that:
$n \ge \sqrt{n+1}+\sqrt{n}$ ?
It should be true for all $n \ge 5$.
Tried it via induction:
*
*$n=5$: $5 \ge \sqrt{5} + \sqrt{6} $ is true.
*$n\implies n+1$:
I need to show that $n+1 \ge \sqrt{n+1} + \sqrt{n+2}$
Starting with $n+1 \ge \sqrt{n} + \sqrt{n+1... | First observe that since $5 \le n$, we have:
$$
4(n+1) =4n+4 < 4n+5 \le 4n+n = 5n \le (n)n = n^2
$$
Hence, since $4(n+1)<n^2 \Rightarrow \boxed{n+1<\dfrac{1}{4}n^2}$ (and $f(x)=\sqrt{x}$ is monotonically increasing and $n\ge0$), we have:
$$
\sqrt{n+1}+\sqrt{n} \le \sqrt{n+1}+\sqrt{n+1} = 2\sqrt{n+1}<2\sqrt{\dfrac{1}{4}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/403090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 6,
"answer_id": 2
} |
If $x\equiv y \pmod{\gcd(a,b)}$, show that there is a unique $z\pmod{\text{lcm}(a,b)}$ with $z\equiv x\pmod a$ and $z\equiv y\pmod b$
If $x\equiv y \pmod{\gcd(a,b)}$, show that there is a unique $z\pmod{\text{lcm}(a,b)}$ with $z\equiv x\pmod{a}$ and $z\equiv y\pmod{b}$
What I have so far:
Let $z \equiv x\pmod{\frac{a... | Existence
Bezout's Identity says that we have $g_a,g_b$ so that
$$
g_aa+g_bb=\gcd(a,b)\tag{1}
$$
Thus,
$$
\begin{align}
g_bb&\equiv\gcd(a,b)&\pmod{a}\\
g_bb&\equiv0&\pmod{b}
\end{align}\tag{2}
$$
Since $x\equiv y\pmod{\gcd(a,b)}$, we have $\gcd(a,b)\mid x{-}y$. Multiplying $(2)$ by $\frac{x-y}{\gcd(a,b)}$ gives
$$
\beg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/404966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Evaluating $\int_{0}^{1} \frac{ \ln x \ln (1-x)}{\sqrt{x} \sqrt{1-x}} dx$ I have the following integral:
$$\int_{0}^{1} \frac{ \ln x \ln (1-x)}{\sqrt{x} \sqrt{1-x}} dx$$
I think I may be to evaluate this with beta and gamma functions but I am not quite sure how. Any help?
| An alternative way, using elementary methods, which begins with the same substitution as sos440's solution:
$$\int_0^1 \frac{\ln x\ln (1-x)}{\sqrt{x(1-x)}}\,dx=8\int_0^{\frac{\pi}{2}}\ln \sin x\ln \cos x\,dx$$
Note that:
$$\begin{aligned}2\int_0^{\frac{\pi}{2}}\ln \sin x\ln \cos x\,dx &=\int_0^{\frac{\pi}{2}}(\ln \sin ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/407304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
} |
Prove inequality: $74 - 37\sqrt 2 \le a+b+6(c+d) \le 74 +37\sqrt 2$ without calculus
Let $a,b,c,d \in \mathbb R$ such that $a^2 + b^2 + 1 = 2(a+b),
c^2 + d^2 + 6^2 = 12(c+d)$, prove inequality without calculus (or
langrange multiplier): $$74 - 37\sqrt 2 \le a+b+6(c+d) \le 74
+37\sqrt 2$$
The original problem is ... | First rewrite the first restriction to $(a-1)^2+(b-1)^2 = 1^2$ and the second restriction to $(c-6)^2+(d-6)^2=6^2$. Hence we may choose $\phi,\theta\in[0,2\pi)$ such that $a = 1+\sin(\phi)$, $b=1+\cos(\phi)$, $c=6+6\sin(\theta)$ and $d = 6+6\cos(\theta)$.
Now $\sin(x)+\cos(x) = \sqrt{2}\sin\left(x+\frac{\pi}{4}\right) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/409979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
How to prove $(a-b)^3 + (b-c)^3 + (c-a)^3 -3(a-b)(b-c)(c-a) = 0$ without calculations I read somewhere that I can prove this identity below with abstract algebra in a simpler and faster way without any calculations, is that true or am I wrong?
$$(a-b)^3 + (b-c)^3 + (c-a)^3 -3(a-b)(b-c)(c-a) = 0$$
Thanks
| Let $ a-b = x $ , $ b -c= y $ and $ c-a =z$ . So actually you want to know the value of $ x^3+y^3+z^3-3xyz$ . Here the technique resides .
$ x+y = (a-c) =-z $ ,
$x^3+y^3+z^3-3xyz = (x+y)^3-3xy(x+y) +z^3-3xyz
= -z^3 +z^3 +3xyz -3xyz = 0 $
Hence the value is $ 0 $
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/413738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 12,
"answer_id": 10
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Probability puzzle - the 3 cannons (Apologies if this is the wrong venue to ask such a question, but I don't understand how to arrive at a solution to this math puzzle).
Three cannons are fighting each other.
Cannon A hits 1/2 of the time. Cannon B hits 1/3 the time. Cannon C hits 1/6 of the time.
Each cannon fires at... | I assume they all fire balls simultaneously. The probability for the different outcomes of the first volley are as then as follows (I will keep my eighteenths unsimplified for simplicity):
1) $A$ is hit: $\frac{1}{3} + \frac{1}{6} - \frac{1}{18} = \frac{8}{18}$
2) $B$ is hit: $\frac{1}{2}$.
And these two are independen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/414334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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Help with a trig-substitution integral I'm in the chapter of trigonometric substitution for integrating different functions. I'm having a bit of trouble even starting this homework question:
$$\int \frac{(x^2+3x+4)\,dx} {\sqrt{x^2-4x}}$$
|
Make the substitution suggested by completing the square, $\sqrt{x^2-4x}=\sqrt{(x-2)^2-4}$, that is, $y=x-2$. Then, by the easily constructed triangle, we have
\begin{align}
\frac{y}{2}
& = \sec\theta\\
\frac{\sqrt{y^2-4}}{2}
& = \tan\theta\\
x^2+3x+4
& =(y+2)^2+3(y+2)+4\\
& = y^2+7y+14\\
\frac{dy}{2}
& = \sec\th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/416088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
} |
Convergence of $\sum_{n=1}^{\infty}\frac{1+\sin^{2}(n)}{3^n}$?
Using either the Direct or Limit Comparison Tests, determine if $\sum_{n=1}^{\infty}\frac{1+\sin^{2}(n)}{3^n}$ is convergent or divergent.
I seem to be completely stuck here.
I've chosen my series to be $\sum\frac{1}{3^n}$, which is clearly a convergent ... | Since $1\le1+\sin^2(n)\le2$, we can compare to the geometric series
$$
\sum_{n=1}^\infty\frac1{3^n}=\frac12\quad\text{and}\quad\sum_{n=1}^\infty\frac2{3^n}=1
$$
In fact, this is actually the sum of three geometric series:
$$
\begin{align}
\sum_{n=1}^\infty\frac{1+\sin^2(n)}{3^n}
&=\sum_{n=1}^\infty\frac{1-\frac14(e^{2i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/419738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Finding all the integer solutions for :$y^2=x^6+17$ Assume that $x,y$ are integers .How to find the solutions for:
$$y^2=x^6+17$$
| $y^2-x^6=17\Rightarrow (y-x^3)(y+x^3)=17$.And we know $17$ is a prime.
So the only possibility is
$y+x^3=17$ and $y-x^3=1$ .
or $y+x^3=-17$ and $y-x^3=-1$
or $y+x^3=1$ and $y-x^3=17$
(As $x,y$ are integers so $(y-x^3)(y+x^3)=17$ implies that the only factors of $17 $ are $y+x^3$ and $y-x^3$ )
$\Rightarrow y=9,x=2$
or... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/423582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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How find this arithmetic sequence of $n$ if there exist positive integer sequence $a_{1},a_{2},a_{3},\cdots,a_{n}$,such that
$$a_{1}a_{2},a_{2}a_{3},a_{3}a_{4},\cdots,a_{n-1}a_{n},a_{n}a_{1}$$ is arithmetic sequence,and the common difference $d=a_{i+1}a_{i+2}-a_{i}a_{i+1}\neq 0,i=1,\cdots,n-1.a_{n+1}=a_{1}$
find the v... | This can be done for any odd $n$. Take $d=n!$ and define $a_1=1\cdot 3 \cdot 5 \cdots$ and $a_2=2 \cdot 4 \cdot 6 \cdots$, in both cases progressing through the odd or even integers up to $n$ until they run out. The sequence of products will be the arithmetic sequence $d,2d,3d,...,nd$ where $nd=n\cdot n!$ The values of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/425130",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
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$a+b+c =0$; $a^2+b^2+c^2=1$. Prove that: $a^2 b^2 c^2 \le \frac{1}{54}$ If a,b,c are real numbers satisfying $a+b+c =0; a^2+b^2+c^2=1$.
Prove that $a^2 b^2 c^2 \le \frac{1}{54}$.
| $c=-a-b$, so
$$a^2+b^2+c^2=2a^2+2ab+2b^2=1$$
let $a=x+y,b=x-y$
then
$$3x^2+y^2=1/2$$
with the we only consider $ab\le 0$
and $c=-(a+b)=-2x$,
$$c^2=a^2+2ab+b^2\le a^2+b^2=1-c^2$$
so $$8x^2\le 1$$
so $$abc=ab(-a-b)=-(a+b)ab=-2x(x^2-y^2)=-x(8x^2-1)$$
then
$$a^2b^2c^2=x^2(8x^2-1)(8x^2-1)=\dfrac{1}{16}\cdot 16x^2(1-8x^2)(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/425187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 8,
"answer_id": 7
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Finding a formula to a given $\sum$ using generating functions Find a close formula to the sum $\sum_{k=0}^{n}k\cdot 5^k$
I tried using generating functions using the differences sequences with no luck.
| Set $g_k=\frac{1}{4}5^{k+1}$. Note that $\Delta g_k=g_{k+1}-g_k=\frac{1}{4}(5^{k+1}-5^k)=5^k$. Set $f_k=k$; we have $\Delta f_k=f_{k+1}-f_k=1$. The partial summation formula gives:
We have $$\sum_0^nk5^k=\sum_0^nf_k\Delta g_k=(f_{n+1}g_{n+1}-f_0g_0)-\sum_0^ng_{k+1}\Delta f_k$$
The RHS simplifies to $$(n+1)\frac{1}{4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/425409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Function problem Show that function $f(x) =\frac{x^2+2x+c}{x^2+4x+3c}$ attains any real value if $0 < c \leq 1$ Problem :
Show that function $f(x)=\dfrac{x^2+2x+c}{x^2+4x+3c}$ attains any real value if $0 < c \leq 1$
My approach :
Let the given function $f(x) =\dfrac{x^2+2x+c}{x^2+4x+3c} = t $ where $t$ is any arbi... | Note that from your quadratic equation in $x$:
$$
(t-1)x^2 + (4t-2)x^2 + (3tc-c) = 0
$$
its discriminant is:
$$
\Delta_x = (4t-2)^2-4(t-1)(3tc-c)=(-12c+16)t^2+(16 c-16)t+(-4c+4)
$$
Note that it suffices to show that $\Delta_x$ is always nonnegative. To do this, we consider its discriminant with respect to $t$:
$$
\Delt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/425573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Algebraic manipulation with square roots I have always had problems with the algebraic manipulation of square roots. For example, recently I encountered this in a problem I was working on:
$$\sqrt{\left(\dfrac{x-1}{2x}\right)^2 - \dfrac{y}{x}} = \dfrac{1}{2x} \sqrt{(x-1)^2 -4xy}$$
I still don't grasp why this is correc... | Observe that:
$$ \begin{align*}
\sqrt{\left(\dfrac{x-1}{2x}\right)^2 - \dfrac{y}{x}}
&= \sqrt{\dfrac{(x-1)^2}{(2x)^2} - \dfrac{y}{x} \cdot \dfrac{4x}{4x}} \\
&= \sqrt{\dfrac{(x-1)^2}{(2x)^2} - \dfrac{4xy}{4x^2}} \\
&= \sqrt{\dfrac{(x-1)^2}{(2x)^2} - \dfrac{4xy}{(2x)^2}} \\
&= \sqrt{\dfrac{(x-1)^2-4xy}{(2x)^2}} \\
&= \s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/425623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Why $x^2 + 7$ is the minimal polynomial for $1 + 2(\zeta + \zeta^2 + \zeta^4)$?
Why $f(x) = x^2 + 7$ is the minimal polynomial for $1 + 2(\zeta + \zeta^2 + \zeta^4)$ (where $\zeta = \zeta_7$ is a primitive root of the unit) over $\mathbb{Q}$?
Of course it's irreducible by the Eisenstein criterion, however it apparent... | Slightly easier:
$(x-1)^2 = 4 ( \zeta^2 + \zeta^4 + \zeta^8 + 2 \zeta^3 + 2 \zeta^5 + 2 \zeta^6) = 4 ( - 2 - \zeta -\zeta^2 - \zeta^4) = 4 ( -2 - \frac{x-1}{2})$
Hence $ x^2 - 2x + 1 = -8 - 2x + 2$ or $x^2 + 7 = 0 $.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/426522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Need help in proving that $\frac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac 1{\sec\theta - \tan\theta}$ We need to prove that $$\dfrac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac 1{\sec\theta - \tan\theta}$$
I have tried and it gets confusing.
| Let $\displaystyle{\sin\theta=\frac{a}{c}}$ and $\displaystyle{\cos\theta=\frac{b}{c}}$ such that $\displaystyle{a^2 + b^2=c^2}$.
$$\displaystyle{\frac{sin\theta - \cos\theta + 1}{\sin\theta+\cos\theta -1}}$$ $$\displaystyle{=\frac{a-b+c}{a+b-c}}$$
Now $\displaystyle{\tan\theta=\frac{a}{b}}$ and $\displaystyle{\sec\th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/426981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 2
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How to find the partial derivative of this complicated function? Find the partial derivative with respect to a (treat other variables as constants):
$\displaystyle f(a,b,c)=\frac {a}{\sqrt{a^2+8bc}}-\frac {a^r}{a^r+b^r+c^r}$
The article I'm reading says it should be:
$\displaystyle \frac {\sqrt{a^2+8bc}-\dfrac{a^2}{\sq... | $$\dfrac{a}{\sqrt{a^2+8bc}}\ge\dfrac{a^r}{a^r+b^r+c^r}\cdots \cdots (1)$$
I guess you do this show that
$$\dfrac{a}{\sqrt{a^2+8bc}}+\dfrac{b}{\sqrt{b^2+8ca}}+\dfrac{c}{\sqrt{c^2+8ab}}\ge 1$$
so you can Substituting $b=c=1$ in (1)
we obtain the following condition on $r:f(a)\ge 0$ for all $a>0$,wher
$$f(a)=a(a^r+2)-a^r\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/428742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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A proposed proof by induction of $1+2+\ldots+n=\frac{n(n+1)}{2}$
Prove: $\displaystyle 1+2+\ldots+n=\frac{n(n+1)}{2}$.
Proof
When $n=1,1=\displaystyle \frac{1(1+1)}{2}$,equality holds.
Suppose when $n=k$, we have $1+2+\dots+k=\frac{k(k+1)}{2}$
When $n = k + 1$:
\begin{align}
1+2+\ldots+k+(k+1) &=\frac{k(k+1)}{2}+k+1 ... | No, all the steps look pretty good; as for the unknown $[\text{step}]$, that is just factoring - although, it is easier to notice that
$$
\frac{k(k+1) + 2k + 2}{2} = \frac{k(k+1) + 2 (k+1)}{2} = \frac{(k+1)(k+2)}{2} .
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/429931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Math Parlor Trick
A magician asks a person in the audience to think of a number $\overline {abc}$. He then asks them to sum up $\overline{acb}, \overline{bac}, \overline{bca}, \overline{cab}, \overline{cba}$ and reveal the result. Suppose it is $3194$. What was the original number?
The obvious approach was modular ar... | The sum of all six combinations is $222(a+b+c)$
So, $3194+100a+10b+c=222(a+b+c)$
As $3194/222>14$
If $a+b+c=15, 100a+10b+c=222(15)-3194=136$
$\implies a+b+c=1+3+6=10\ne 15$
If $a+b+c=16, 100a+10b+c=222(16)-3194=358$
$\implies a+b+c=3+5+8=16$ as needed
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/432118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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How to integrate these integrals $$\int^{\frac {\pi}2}_0 \frac {dx}{1+ \cos x}$$
$$\int^{\frac {\pi}2}_0 \frac {dx}{1+ \sin x}$$
It seems that substitutions make things worse:
$$\int \frac {dx}{1+ \cos x} ; t = 1 + \cos x; dt = -\sin x dx ; \sin x = \sqrt{1 - \cos^2 x} = \sqrt{1 - (t-1)^2} $$
$$ \Rightarrow
\int \frac ... | $\int_a^bf(x)dx=\int_a^bf(a+b-x)dx$
Therefore, $$\int_0^{\pi/2}\dfrac{1}{1+\cos x}dx=\int_0^{\pi/2}\dfrac{1}{1+\sin x}dx$$
As you did $$\int_0^{\pi/2}\dfrac{1}{1+\cos x}dx=\frac{1}{2}\int_0^{\pi/2}\sec^2\frac{x}{2}dx$$
Substitute $x=2t$ and use $\int\sec^2 tdt=\tan t+C$
EDIT: Proof of $\int_a^bf(x)dx=\int_a^bf(a+b-x)dx... | {
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"question_score": "2",
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"answer_id": 4
} |
simple limit but I forget how to prove it I have to calculate the following limit
$$\lim_{x\rightarrow -\infty} \sqrt{x^2+2x+2} - x$$
it is in un undeterminated form.
I tried to rewrite it as follows:
$$\lim_{x\rightarrow -\infty} \sqrt{x^2+2x+2} - \sqrt{|x|^2}$$
but seems a dead road.
Can anyone suggest a solution?
... | HINT:
Putting $y=-x,$
$$ \sqrt{x^2+2x+2} - x=\sqrt{y^2-2y+2} +y= \frac{(\sqrt{y^2-2y+2} +y)(\sqrt{y^2-2y+2} -y)}{(\sqrt{y^2-2y+2} -y)}$$
$$=\frac{y^2-2y+2-y^2}{\sqrt{y^2-2y+2} -y}=\frac{-2+\frac2y}{\sqrt{1-\frac2y+\frac2{y^2}}-1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/434370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Question about Permutations, and the distinct differences I have the following question regarding permutations of the sequence $(1,2,\cdots,n)$:
For what values of $n$ does there exist a permutation $(x_1,x_2,\cdots,x_n)$ of $(1,2,\cdots,n)$, such that the differences $|x_k-k|$ for each $k\in\{1,2,\cdots,n\}$ are all d... | I'm not sure, but I think I've found a construction for $n=0\mod 4$. First, some examples:
$$\begin{pmatrix}1 & 4 & 2 & 3\\
4 & 2 & 1 & 3
\end{pmatrix} \\
\begin{pmatrix}1 & 8 & 2 & 7 & 3 & 5 & 4 & 6\\
8 & 2 & 7 & 3 & 5 & 4 & 1 & 6
\end{pmatrix} \\
\begin{pmatrix}1 & 12 & 2 & 11 & 3 & 10 & 4 & 8 & 5 & 7 & 6 & 9\\
12 & ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/435439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
How can I show that $\sqrt{1+\sqrt{2+\sqrt{3+\sqrt\ldots}}}$ exists? I would like to investigate the convergence of
$$\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\sqrt\ldots}}}}$$
Or more precisely, let $$\begin{align}
a_1 & = \sqrt 1\\
a_2 & = \sqrt{1+\sqrt2}\\
a_3 & = \sqrt{1+\sqrt{2+\sqrt 3}}\\
a_4 & = \sqrt{1+\sqrt{2+\sqrt{3+\... | The proofs previous to this one have all been non-constructive. They all state an upper bound, observe that the sequence is monotonically increasing, and then appeal to the Monotone Convergence Theorem. This type of argument cannot provide a computable error bound. The following does.
Call the expression $\sqrt{1 + \sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/437209",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "63",
"answer_count": 7,
"answer_id": 2
} |
convergence to a generalized Euler constant and relation to Zeta serie Let $0 \leq a \leq 1$ be a real number. I would like to know how to prove that the following sequence converges:
$$u_n(a)=\sum_{k=1}^n k^a- n^a \left(\frac{n}{1+a}+\frac{1}{2}\right)$$
For $a=1$:
$$u_n(1)=\sum\limits_{k=1}^{n} k- n \left(\frac{n}{1+... | To avoid a black box formula
Since the function $x\mapsto x^a$ is increasing then we find easily the asymptotic expansion
$$\sum_{k=1}^n k^a\sim_\infty \int_1^n x^a dx\sim_\infty \frac{1}{a+1}n^{a+1}\tag{AE}$$
hence we have
$$\sum_{k=1}^n k^a=\frac{1}{a+1}n^{a+1}+O\left(n^{a}\right)\tag{1}$$
Now let's improve the equ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/437507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Minimum and maximum of $ \sin^2(\sin x) + \cos^2(\cos x) $ I want to find the maximum and minimum value of this expression:
$$ \sin^2(\sin x) + \cos^2(\cos x) $$
| Following George's hint,
Because $-1\le \sin x \le 1$, and $\sin x$ is strictly increasing on $-1\le x\le 1$, we see that $\sin (\sin x)$ (and hence $\sin^2(\sin x)$) is maximized when $\sin x=1$, e.g. at $x=\pi/2$.
On the other hand, $\cos x$ is maximized when $x=0$, so $\cos (\cos x)$ (and hence $\cos^2(\cos x)$ is m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/437556",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Recursion Question - Trying to understand the concept Just trying to grasp this concept and was hoping someone could help me a bit. I am taking a discrete math class. Can someone please explain this equation to me a bit?
$f(0) = 3$
$f(n+1) = 2f(n) + 3$
$f(1) = 2f(0) + 3 = 2 \cdot 3 + 3 = 9$
$f(2) = 2f(1) + 3 = 2 \cdo... | The function $f$ is given by a starting condition $f(0)=3$, and by the condition $f(n+1)=2\cdot f(n)+3$. This means that we calculate the value of $f(2)$ from the value of $f(1)$, which itself is calculated from $f(0)$ -- which was given to us.
$$f(1)=9\implies f(2)=2\cdot f(1)+3=2\cdot 9+3$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/441718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Discrete Math Question: arithmetic progression A lumberjack has $4n + 110$ logs in a pile consisting of n layers. Each layer has two more logs than the layer directly above it. If the top layer has six logs, how many layers are there? Write the steps to calculate the equation for the problem and state the number of lay... | As you observe, you have an arithmetic progression, and your work is fine to the point you left off; solving for $n$ is mostly a matter of algebra:
Picking up where you left off
$$\begin{align} 4n + 110 & = \dfrac n2\left(6 + 6 + 2(n-1)\right) \\ \\
& = \dfrac n2(12) + \dfrac {2n\cdot (n-1)}{2}\\ \\
& = 6n + n(n - 1) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/443079",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
What is the derivative of $x\sin x$? Ok so I know the answer of $\frac{d}{dx}x\sin(x) = \sin(x)+ x\cos(x)$...but how exactly do you get there? I know $\frac{d}{dx} \sin{x} = \cos{x}$. But where does the additional $\sin(x)$ (in the answer) come in?
| Depending on how familiar you are with the chain rule and derivative of logarithmic functions, you could differentiate $x \sin{x}$ using this alternate method that doesn't directly use the product rule,
$$\begin{align} y &= x \sin{x} \\
\ln(y) &= \ln( x \sin{x}) \\
\ln(y) &= \ln(x) + \ln(\sin{x}) \\
\frac{1}{y} \frac{d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/443509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Find y'' using implicit differentiation Pretty sure I messed up
$$
x^3 - 3xy + y^3 = 1
$$
$$
3x^2-3xy'-3y+3y^2y'=0
$$
$$
3y^2y'-3xy'=3y-3x^2
$$
$$
(3y^2-3x)y'=3y-3x^2
$$
$$
y' = \dfrac{y-x^2}{y^2-x}
$$
$$
y'' = \dfrac{(y^2-x)(y'-2x)-(y-x^2)(2yy'-1)}{(y^2-x)^2}
$$
$$
y'' = \dfrac{y^2y'-2xy^2-xy'+2x^2-2y^2y'+y+2x^2yy'-x^... | As pointed out in the comments, the error occurred when plugging back in the value for $y'$. The part that reads:
$$x^2-2xy^2+y$$
Should read:
$$(x^2-2xy^2+y)(y^2-x)$$
This multiplies out and helps simplify further.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/445448",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Are there $a,b>1$ with $a^4\equiv 1 \pmod{b^2}$ and $b^4\equiv1 \pmod{a^2}$? Are there solutions in integers $a,b>1$ to the following simultaneous congruences?
$$
a^4\equiv 1 \pmod{b^2} \quad \mathrm{and} \quad b^4\equiv1 \pmod{a^2}
$$
A brute-force search didn't turn up any small ones, but I also don't see how to rule... | I would think this could be attacked as follows. Write $b = b_1b_2b_3$ for integers $b_1,b_2,b_3$, such that $b_1^2 \mid (a-1)$ and $b_2^2 \mid (a+1)$ and $b_3^2 \mid (a^2+1)$. Hence the first condition is satisfied, i.e.,
$$ a^4-1 = (a-1)(a+1)(a^2+1) \equiv 0 \pmod{b^2}.$$
Write $a-1 = c_1b_1^2$ and $a+1=c_2b_2^2$ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/446269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 2,
"answer_id": 1
} |
Big $\Omega$ question! Prove $(n-1)(n-2)(n-3)$ is $\Omega(n^3)$ Problem
Prove $(n-1)(n-2)(n-3)$ is $\Omega(n^3)$.
Attempt @ Solution
*
*$f(n) = n^3(1-6/n+11/n^2-6/n^3)$
*$g(n) = n^3$
*Show that there exists a $C > 0$ and $n_0$ such that $f(n) \ge Cg(n)$ for all $n > n_0$.
*I tried plugging in different numbers fo... | Your claim is that $n_0=7$ and $C=1$ work: that if $n>7$, then $f(n)\ge g(n)$, but you haven’t actually proved it. And you cannot possible prove it, because
$$\frac{f(n)}{g(n)}=\frac{(n-1)(n-2)(n-3)}{n^3}=\frac{n-1}n\cdot\frac{n-2}n\cdot\frac{n-3}n<1^3=1\tag{1}$$
for all $n\ge 1$, and therefore $f(n)<g(n)$ for all $n\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/447773",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
$6$ people, $3$ rooms, $1$ opening door $6$ people spread in $3$ distinguishable rooms, every room needs one person who opens the door.
There are ${6 \choose 3}\cdot 3 \cdot 2$ options to choose the three door opener persons and let them open one certain room, so that every room is opened by one person.
Further, there ... | Here's an easy way to look at the problem:
First, we make sure that we have a door opener in each room. There are $6$ choices for the first room, $5$ for the second, and $4$ for the third. Thus, we may select door openers in $6\times5\times4=120$ ways.
Now, each of the three remaining people can be put in any of the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/449637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
A problem about the convergence of an improper integral Let $f:\mathbb R\longrightarrow\mathbb R$ be a function with
$$f(x)=\frac{1}{3}\int_x^{x+3} e^{-t^2}dt$$
and consider $g(x)=x^nf(x)$ where $n\in\mathbb Z$.
I have to discuss the convergence of the integral
$$\int_{-\infty}^{+\infty}g(x)dx$$ at the varying o... | I think I can give you expression for $\int_{-\infty}^{\infty}g(x)\ dx$. Then, we can think about how it should converge.
We start by noting that $$\large f(x)=\frac{\sqrt{\pi}}{3}\left(Q\left(\sqrt{2}x\right)-Q\left(\sqrt{2}(x+3)\right)\right)$$ and $$\large f(-x)=\frac{\sqrt{\pi}}{3}\left(-Q\left(\sqrt{2}x\right)+Q\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/449813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 3
} |
What is the number of real solutions of the following? $ \sqrt{x + 3 - 4\sqrt{x-1}} + \sqrt{x + 8 - 6\sqrt{x-1}} = 1 $ What is the number of real solutions of the following?
$$ \sqrt{x + 3 - 4\sqrt{x-1}} + \sqrt{x + 8 - 6\sqrt{x-1}} = 1 $$
My solution:
$$ \sqrt{x + 3 - 4\sqrt{x-1}} + \sqrt{x + 8 - 6\sqrt{x-1}} = 1 $$
... | It happens, for the choices of the arguments of the radicals in this problem, that we can make this a bit less of a headache to think about by noting that setting $ \ u = \sqrt{x-1} \ \Rightarrow \ u^2 = x - 1 \ $ reduces the original equation to
$$ \sqrt{(u^2 + 4) - 4u} \ + \ \sqrt{(u^2 + 9) - 6u} \ = \ 1 \ \Rightar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/452927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
If $(2x^2-3x+1)(2x^2+5x+1)=9x^2$,then prove that the equation has real roots. If $(2x^2-3x+1)(2x^2+5x+1)=9x^2$,then prove that the equation has real roots.
MY attempt:
We can open and get a bi quadratic but that is two difficult to show that it has real roots.THere must be an easy way.!
| First divide both members of the equation by $x^2$:
$$(2x + \frac{1}{x} - 3)(2x + \frac{1}{x} + 5) = 9$$
With the notation $2x + \frac{1}{x} = y$ equation is obtained in $y$: $$y^2 + 2y - 24 =0$$ with roots $y_1 = -6$ and $y_2=4$.
The equation $$2x + \frac{1}{x} = -6$$ has roots $x_1,_2 = \frac{-3\pm\sqrt{7}}{2}$;
Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/453941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
$24\mid n(n^{2}-1)(3n+2)$ for all $n$ natural problems in the statement. "Prove that for every $ n $ natural, $24\mid n(n^2-1)(3n+2)$"
Resolution:
$$24\mid n(n^2-1)(3n+2)$$if$$3\cdot8\mid n(n^2-1)(3n+2)$$since$$n(n^2-1)(3n+2)=(n-1)n(n+1)(3n+2)\Rightarrow3\mid n(n^{2}-1)(3n+2)$$and$$8\mid n(n^{2}-1)(3n+2)?$$$$$$
Would n... | We know from here or here, the product $n$ consecutive integers is divisible by $n!$ for integer $n>0$
As $24=4!$and we already have $n(n^2-1)=(n-1)n(n+1)$ the product of $3$ consecutive integers
So, if we can arrange the next or previous term as multiplier, we shall have the product of $4$ consecutive integers, henc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/455043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Is $83^{27} +1 $ a prime number? I'm having problems with exercises on proving whether or not a given number is prime. Is $83^{27} + 1$ prime?
| We have a chain of divisibilities, based on the fact that $(a-b)\mid(a^n-b^n)$,
$$
83^1-(-1)^1\mid83^3-(-1)^3\mid83^9-(-1)^9\mid83^{27}-(-1)^{27}=83^{27}+1
$$
Using this chain, we get, using $a^3-b^3=(a-b)(a^2+ab+b^2)$,
$$
\begin{align}
83^{27}+1
&=\frac{83^{27}+1}{83^9+1}\times\frac{83^9+1}{83^3+1}\times\frac{83^3+1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/458323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 9,
"answer_id": 7
} |
$(a^{n},b^{n})=(a,b)^{n}$ and $[a^{n},b^{n}]=[a,b]^{n}$? How to show that $$(a^{n},b^{n})=(a,b)^{n}$$ and $$[a^{n},b^{n}]=[a,b]^{n}$$ without using modular arithmetic? Seems to have very interesting applications.$$$$Try: $(a^{n},b^{n})=d\Longrightarrow d\mid a^{n}$ and $d\mid b^n$
| Bezout's Identity says that $\gcd(a,b)$ is the smallest positive element of $\{ax+by:x,y\in\mathbb{Z}\}$.
The smallest positive element of $\{acx+bcy:x,y\in\mathbb{Z}\}$ is $\gcd(ac,bc)$; it is also $c$ times the smallest positive element of $\{ax+by:x,y\in\mathbb{Z}\}$ which is $\gcd(a,b)$. Therefore
$$
\gcd(ac,bc)=c\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/460231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Quadratic inequality with boundaries Here is a very old high school exam question I am trying to solve (purely for interest only):
If $a,b,c$ are real numbers such that $-1 \le ax^2+bx+c \le 1$ for $-1 \le x \le 1$ prove that $-4 \le 2ax+b \le4$ for $-1 \le x \le 1$ (Hint: Consider the functions at the end-points and a... | We can show the required inequlity in an entirely algebraic manner:
We first note that by setting $x=0$ in our initial inequality we get:
$$-1\le c\le1 \\ |c|\le1.$$
If we set $x=\pm1$ in our given inequality we will get:
$$
\text{(i)}\;\;-1\le a+b+c\le 1 \;\;\;\text{and}\;\;\;\text{(ii)}\;\;-1\le a-b+c\le 1.
$$
Addin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/460889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Positive semi-definite matrix problem If $A，B$ and $M$ are positive semi-definite matrices, and we have
$$ A+B \succeq M .$$
Do there always exist two positive semi-definite matrices $ M_{1}, M_{2} , $ such that
$$ A \succeq M_{1}, \quad B \succeq M_{2}, \quad M_{1}+M_{2}=M ?$$
| The statement is false even when we impose stronger conditions of positive definiteness on the matrices.
Let $p > 0$ be a number to be determined. Consider the following martices:
$$
A = \begin{pmatrix}2p + 1& 0\\0&1\end{pmatrix}, \quad
B = \begin{pmatrix}1 & 0\\0 & 2p + 1\end{pmatrix} \quad\text{ and }\quad
M = \begin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/461694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
How do I show that $\gcd(a^2, b^2) = 1$ when $\gcd(a,b)=1$? How do I show that $\gcd(a^2, b^2) = 1$ when $\gcd(a,b)=1$?
I can show that $\gcd(a,b)=1$ implies $\gcd(a^2,b)=1$ and $\gcd(a,b^2)=1$. But what do I do here?
| If $\gcd(a, b) = 1$, this means exactly that $a$ and $b$ do not have any prime factors in common. Any common factor would constitute a common divisor. If $a$ and $b$ have some prime factors in common, then the product of those common factors (including any repeated ones of higher multiplicity) is the greatest common di... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/463194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
What is the sum of this by telescopic method? $$\sum_{r=1}^\infty \frac{r^3+(r^2+1)^2}{(r^4+r^2+1)(r^2+r)}$$
This is to be done by telescopic method. I've used many things like writing $r^4+r^2+1=(r^2+r+1)(r^2-r+1)$ but have failed. I think all it needs is the correct rearranging of terms.
| Hints:
*
*Since $r^3+(r^2+1)^2=r(r^2+r)+(r^4+r^2+1)$,
$$
\frac{r^3+(r^2+1)^2}{(r^4+r^2+1)(r^2+r)}=\frac{r}{r^4+r^2+1}+\frac1{r^2+r}.
$$
*Since $r^4+r^2+1=(r^2+r+1)(r^2-r+1)$,
$$
\frac{2r}{r^4+r^2+1}=a_r-a_{r+1},\qquad a_r=\frac{1}{r(r-1)+1}.
$$
*Since $r^2+r=r(r+1)$,
$$
\frac1{r^2+r}=b_r-b_{r+1},\qquad b_r=\frac1r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/464417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Given $\sec \theta + \tan \theta = 5$ , Find $\csc \theta + \cot \theta $. The question is to find the value of $ \csc \theta + \cot \theta $ if $\sec \theta + \tan \theta = 5$ .
Here is what I did :
$\sec \theta + \tan \theta = 5$
$\sec \theta = 5 - \tan \theta $
Squaring both sides ,
$$\sec^2 \theta = 25 + \... | Too long for a comment:
...
THIS IS HOW I THINK THE ANSWER SHEET WRITER DID THE MISTAKE:
Normally the equation is
$$\sin\theta+1=5\cos\theta.$$
Squaring they "mistakenly" got $\sin^2\theta+2\sin\theta+1=5\cos^2\theta\implies$ $\sin^2\theta+2\sin\theta+1=5-5\sin^2\theta\implies$ $3\sin^2\theta+\sin\theta-2=0\implies$ $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/464472",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 3
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solve differential equation using given substitution Solve the following equation by making substitution
$$y=xz^n \text{ or } z=\frac{y}{x^n}$$
and choosing a convenient value of n.
$$\frac{dy}{dx}= \frac{2y}{x} +\frac{x^3}{y} +x\tan\frac{y}{x^2}$$
I thought it can be solved in 2 ways
1.making it exact differential equ... | Since the ODE has the term $x\tan\dfrac{y}{x^2}$ present, so it is obviously to choose $n=2$ .
Let $z=\dfrac{y}{x^2}$ ,
Then $y=x^2z$
$\dfrac{dy}{dx}=x^2\dfrac{dz}{dx}+2xz$
$\therefore x^2\dfrac{dz}{dx}+2xz=2xz+\dfrac{x}{z}+x\tan z$
$x^2\dfrac{dz}{dx}=x\left(\dfrac{1}{z}+\tan z\right)$
$\dfrac{dx}{x}=\dfrac{z}{z\tan z+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/464818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Area ratio in triangle? Given: $\triangle ABC$. In the side $AB$, we choose point $D$. From this point $D$, we draw a line $DF$ such that intersect side $AC$ and line $DE$ such that intersect side $BC$. If $DF\parallel BC$, $DE\parallel AC$, and the area of $\triangle BDE = p$ times area of $\triangle$ADF, what is the ... | The quadrilateral $CEDF$ is a parallelogram, and therefore $CE=DF=d$, $CF=DE=e$, and $\triangle CEF \equiv \triangle DEF$.
Suppose $h$ is the height of parallelogram $CEDF$ and $\ell$ is the height of $\triangle ADF$. Then, $h$ will also be the height of $\triangle BDE$ and $\triangle CEF$.
$$\therefore \ \text{Area o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/467469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find the 12th term and the sum of the first 12 terms of a geometric sequence. A geometric series has a first term $\sqrt{2}$ and a second term $\sqrt{6}$ . Find the 12th term and the sum of the first 12 terms.
I can get to the answers as irrational numbers using a calculator but how can I can obtain the two answers in ... | The general term of a geometric series is $a r^{n-1}$, where $r$ is the ratio, and $a$ is the first term. In your case, $a=\sqrt{2}$ and $r=\sqrt{3}$. The 12th term is then
$$\sqrt{2} (\sqrt{3})^{11} = 243 \sqrt{6}$$
The sum of the first 12 terms is
$$a \sum_{k=0}^{11} r^k = a \frac{r^{12}-1}{r-1} = \sqrt{2} \frac{72... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/468331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Evaluating $\int_0^{\infty} {y^2 \cos^2(\frac{\pi y}{2}) \over (y^2-1)^2} dy$ I´m having trouble with the following integral
$$
\int_0^{\infty} {y^2 \cos^2(\frac{\pi y}{2}) \over (y^2-1)^2} dy
$$
I have tried lots of approaches and nothing works. Mathematica says it does not converge but that is not true. It appears in... | Let $I$ denote the integral. Then
\begin{align*}
I &= \frac{1}{2} \int_{-\infty}^{\infty} \frac{y^2 \cos^{2} (\pi y/2)}{(y^{2} - 1)^{2}} \, dx \\
&= \frac{1}{4} \int_{-\infty}^{\infty} \frac{y^2 (1 + \cos \pi y)}{(y^{2} - 1)^{2}} \, dx \\
&= \frac{1}{4} \Re \mathrm{PV}\!\!\int_{-\infty}^{\infty} \frac{y^2 (1 + e^{i\pi ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/468664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Proof: for a pretty nasty limit Let $$ f(x) = \lim_{n\to\infty} \dfrac {[(1x)^2]+[(2x)^2]+\ldots+[(nx)^2]} {n^3}$$.
Prove that f(x) is continuous function.
Edit: $[.] $ is the greatest integer function.
| $k^2x^2-1 < [(kx)^2] \le k^2x^2$
Summing the three expressions from $k=1$ to $n$ and dividing by $n^3$, we get
$\frac{x^2}{6}(1+\frac{1}{n})(2+\frac{1}{n}) - \frac{1}{n^2}< \frac{[(1x)^2] + \cdots + [(nx)^2]}{n^3} \le \frac{x^2}{6}(1+\frac{1}{n})(2+\frac{1}{n}) $.
Taking limits as $n \rightarrow \infty$, we get
$\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/470247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Solving inequalities, simplifying radicals, and factoring. (Pre calculus) (Q.1) Solve for $x$ in $x^3 - 5x > 4x^2$
its a question in pre calculus for dummies workbook, chapter 2. The answer says:
then factor the quadratic: $x(x-5)(x+1)>0$. Set your factors equal to $0$ so you can find your key points.When you have them... | multiply the numerator AND denominator by one more cube root of 4.
$$\frac{8}{4^{\frac{2}{3}}} = \frac{8}{\sqrt[3]{4^2}}\cdot \frac{\sqrt[3]{4^1}}{\sqrt[3]{4^1}} = \frac{8\cdot\sqrt[3]4}{\sqrt[3]{4^3}}=\frac{8\cdot\sqrt[3]4}{4}=2\cdot\sqrt[3]4$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/471807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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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.