Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
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Prove that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \frac{9}{a+b+c} : (a, b, c) > 0$ Please help me for prove this inequality:
$$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \frac{9}{a+b+c} : (a, b, c) > 0$$
| There is a basic inequality you should know:
$$(a_{1}+a_{2}+\cdots+a_{n}) \cdot \left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots +\frac{1}{a_{n}}\right) \ge n^2 $$
that is elementarily solved by Cauchy-Schwarz in a single row proof
$$\left(\sqrt{a_{1}}\frac{1}{\sqrt{a_{1}}}+\cdots +\sqrt{a_{n}}\frac{1}{\sqrt{a_{n}}}\right... | {
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"timestamp": "2023-03-29T00:00:00",
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Why does $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$? Playing around on wolframalpha shows $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$. I know $\tan^{-1}(1)=\pi/4$, but how could you compute that $\tan^{-1}(2)+\tan^{-1}(3)=\frac{3}{4}\pi$ to get this result?
| $$\tan^{-1}(2)+\tan^{-1}(3)=\tan^{-1}{\left(\frac{2+3}{1-2\cdot 3}\right)}=\tan^{-1}(-1)=n\pi-\frac \pi 4,$$ where $n$ is any integer.
Now the principal value of $\tan^{-1}(x)$ lies in $[-\frac \pi 2, \frac \pi 2]$ precisely in $(0, \frac \pi 2)$ if finite $x>0$.
So, the principal value of $\tan^{-1}(2)+\tan^{-1}(3)$ w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/197393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is it possible to take the absolute value of both sides of an equation? I have a problem that says: Suppose $3x^2+bx+7 > 0$ for every number $x$, Show that $|b|<2\sqrt21$.
Since the quadratic is greater than 0, I assume that there are no real solutions since
$y = 3x^2+bx+7$, and $3x^2+bx+7 > 0$, $y > 0$
since $y>0$ ... | From $b^2\lt 84$, you cannot conclude that $b\lt \pm\sqrt{84}$, whatever that may mean. It cannot mean that $b\lt \sqrt{84}$ or $b\lt -\sqrt{84}$, since $-100$ is less than each of $\sqrt{84}$ and $-\sqrt{84}$.
What you probably intend to say is that $b^2\lt 84$ iff $-\sqrt{84}\lt b\lt \sqrt{84}$. And we can rewrite t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/200946",
"timestamp": "2023-03-29T00:00:00",
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Sum and Product of Infinite Radicals How to compute $$\sqrt{(1\sqrt{(2\sqrt{(3\dots)})})}$$ & $$\sqrt{(1+\sqrt{(2+\sqrt{(3+\cdots)})})}$$?
I understand that $$\sqrt{(1\sqrt{(2\sqrt{(3\dots)})})}=(1^{1/2})(2^{1/4})(3^{1/8})\cdots$$
and
$$\sqrt{(1+\sqrt{(2+\sqrt{(3+\cdots)})})}=(1+(2+(3+(\cdots))^{1/8})^{1/4})^{1/2} $$
... | About
$$ c=\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\ldots}}}} $$
we may notice that for any $n\geq 1$ the functions $f_n(x)=\sqrt{n+x}$ are contractions of $[\varphi,+\infty)$ with Lipschitz constant $\leq\frac{1}{2\sqrt{n+1}}$. Additionally, for any $n\geq 2$ we have
$$ \sqrt{n+\sqrt{n+1+\sqrt{n+2+\sqrt{n+3+\ldots}}}}<\sqrt{n... | {
"language": "en",
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Explain why 67 is prime based on the fact that order of 2 mod 67 is 66 Without using the fact that 67 is prime, show that the order of 2 mod 67 is 66. Explain why this result proves that 67 is prime
What I understand:
*
*The order of 2 in $\mathbb{Z}_{67}$(or mod $67$) $ = 66$ means that $66$ is the smallest power $... | $2^6\equiv64\pmod {67} \implies 2^6\equiv -3 \pmod {67}\implies {(2^6)}^{11}\equiv -3^{11}\pmod {67}$
Now, $3^4\equiv 14\pmod {67}\implies 3^8\equiv 196\pmod {67}\equiv{-5}\pmod {67}$
$\implies 3^{11}=3^8.3^3\equiv -135\pmod {67}$
Therefore, $2^{66}\pmod {67}\equiv-3^{11}\pmod {67}\equiv 135\pmod {67}\equiv 1\pmod {67}... | {
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For this non-linear differential equation: $ xy' = y + ax \sqrt{x^2+y^2} $, where $ x>0 $ and $ a>0$ is a constant This is a 3 part question (I seem to have found (a), but I am not confident about (b) and thus (c) as well):
(a) if y is a solution, show that v:= $yx^{-1}$ satisfies the differential equation:
$v'=a \sqr... | After getting differential eqn $$\frac{dv}{dx}=a\sqrt{1+v^2}$$ $$\implies \int\frac{dv}{\sqrt{1+v^2}} =\int a dx$$ $$\implies \ln(v+\sqrt{1+v^2})=ax+c$$
Putting $v=\frac{y}{x}$ gives
$$\ln(\frac{y+\sqrt{x^2+y^2}}{x})=ax+c$$
Since $y(1)=0\implies \ln(1)=a+c\implies c=-a$ (as $\ln(1 )=0$)
Thus, particular solution is $... | {
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Proof that $\sqrt[3]{3}$ is irrational Proof that $\sqrt[3]{3}$ (Just need a check)
Let $\sqrt[3]{3}= \frac{a}{b}$ both a,b are integers of course. $\Rightarrow 3=\frac{a^{3}}{b^{3}}$ $\Rightarrow$ $3b^{3}=a^{3}$ $\Rightarrow$ 3b=a $\Rightarrow$ $3b^{3}=27b^{3}$ and this is a contradiction because the cubing function i... | Supposing $\sqrt[3]{3}$ is a rational solution of an equation, we can assume $x=\sqrt[3]{3}$ and $x^3=3$.
But $x^3=3 \Rightarrow x^3-3 =0$. The rational-root theorem tells us that for a polynomial $f(x) = a_nx^n + a_{n−1}x^{n−1} + \ldots + a_1x + a_0$ all of whose coefficients ($a_n$ through $a_0$) are integers, the re... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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"answer_id": 3
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probability word problems with marbles Select 3 marbles at random from a jar with 6-blue, 4-green, and 3-red. What is probability that
A.All are green?
B.What is probability of selecting 2-blue and 1-red?
C.What is probability of selecting in exact order 1 blue, 1 green, and 1 red?
| There are $13$ balls, so $\dbinom{13}{3}$ ways to choose $3$ balls. All these ways are equally likely. There are $\dbinom{4}{3}$ ways to choose $3$ green. Divide.
B: Same denominator. There are $\dbinom{6}{2}$ ways to choose $2$ blue. For each of these ways, there are $\dbinom{3}{1}$ ways to choose $1$ red, for a total... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/204280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Central Projection of the Sphere
Given the parameterization of the unit sphere $x^2+y^2+z^2=1$ as
$x = \displaystyle\frac{u}{\sqrt{1+u^2+v^2}} $
$y = \displaystyle\frac{v}{\sqrt{1+u^2+v^2}} $
$z = \displaystyle\frac{1}{\sqrt{1+u^2+v^2}} $
Find $ds^2=dx^2+dy^2+dz^2$ and using the metric computer the area of
the hemi... | Factor your coefficients:
$$
v^4 + 2 v^2 + 1 + u^2 + u^2 v^2 = (1+v^2)(1+u^2+v^2)
$$
$$
u^4 + 2 u^2 + 1 + v^2 + u^2 v^2 = (1+u^2) (1+u^2+v^2)
$$
$$
(-2 u v-2 u^2 v-2 u v^3) = - 2u v(1+u^2+v^2)
$$
Then cancel common factors of the numerator and the denominator.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How $v=(-\sin\frac{\alpha}{2}\sin\frac{\beta}{2},\cos\frac{\alpha}{2}\sin\frac{\beta}{2},\sin\frac{\alpha}{2}\cos\frac{\beta}{2})$ is derived from... In the book Quarternion and Rotation Sequences, I can't seem to work out how the final equation (colored in $\color{red}{red}$) is derived from the original equation (col... | Facts:
*
*$1-\cos{2\theta}=2\sin^2\frac{\theta}{2}$.
*$\sin{\theta}=2\sin\frac{\theta}{2}\cos\frac{\theta}{2}$, which we can rearrange to $\displaystyle\frac{\sin{\theta}}{2\sin\frac{\theta}{2}}=\cos\frac{\theta}{2}$.
Steps:
So, starting with $$\color{blue}{\vec{v}=\left(k,\frac{k\sin\alpha}{\cos\alpha-1},\frac{k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/208561",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving $\sum\limits_{i=1}^{n}{\frac{i}{2^i}}=2-\frac{n+2}{2^n}$ by induction.
Theorem (Principle of mathematical induction):
Let $G\subseteq \mathbb{N}$, suppose that
a. $1\in G$
b. if $n\in \mathbb{N}$ and $\{1,...,n\}\subseteq G$, then $n+1\in G$
Then $G=\mathbb{N}$
Proof by Induction
Prove that
... | Absolutely correct, this is it.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Proving that a set is not an ordered field Problem:
Let $S=\left \{ 0,1,2 \right \}$. How can someone prove that there is unique way of defining addition and multiplication such that $S$ is a field if $0$ of the set $S$ has the meaning (for any element $a$ in $S$: $0+a=a$), and $1$ in $S$ has the meaning (for any eleme... | The axioms for a ring already imply that $0\cdot a=0$ for all $a\in S$, so you have almost all of the multiplication table:
$$\begin{array}{c|cc}
\cdot&0&1&2\\ \hline
0&0&0&0\\
1&0&1&2\\
2&0&2
\end{array}$$
And $2$ has to have a multiplicative inverse, so we must have $2\cdot 2=1$.
For addition we automatically have th... | {
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"url": "https://math.stackexchange.com/questions/210260",
"timestamp": "2023-03-29T00:00:00",
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Solving trigonometric equations of the form $a\sin x + b\cos x = c$ Suppose that there is a trigonometric equation of the form $a\sin x + b\cos x = c$, where $a,b,c$ are real and $0 < x < 2\pi$. An example equation would go the following: $\sqrt{3}\sin x + \cos x = 2$ where $0<x<2\pi$.
How do you solve this equation wi... | The idea is to use the identity $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$. You have $a\sin x+b\cos x$, so you’d like to find an angle $\beta$ such that $\cos\beta=a$ and $\sin\beta=b$, for then you could write
$$a\sin x+b\cos x=\cos\beta\sin x+\sin\beta\cos x=\sin(x+\beta)\;.$$
The problem is that $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/213545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solve three simultaneous equations with three unknowns $0 = 9a + 3b + c$
$2 = 25a + 5b + c$
$6 = 49a + 7b + c$
These are my 3 equations that have 3 unknowns. How do I solve for these unknowns?
| (1) $9a+3b+c=0$
(2) $25a+5b+c=2$
(3) $49a+7b+c=6$
Using (2)-(1) we have:
(4) $16a+2b=2$
Using (3)-(2) we have:
(5) $24a+2b=4$
Using (5)-(4) we have:
(6) $8a=2$
It follows that $a=\frac{1}{4}$, $b=-1$ and $c=\frac{3}{4}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/216593",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Urgent - Find the equation of the lines tangent to a circle Question: 'Find the equation of the lines from point $P(0,6)$ tangent to the circle $x^2+y^2=4x+4$.
So what I did firstly is rewrite it to the form $(x-2)^2 + y^2 = 8$, and I saw that point $P$ is not on the circle. I learned that the equation of the line tang... | The equation of any line passing through $(0,6)$ can be written as $\frac{y-6}{x-0}=m$ where $m$ is the gradient, So, $y=mx+6$
If the $(h,k)$ be the point of contact, then $k=mh+6$ and $h^2+k^2-4h-4=0$
Replacing $k$ in the 2nd equation, $h^2+(mh+6)^2-4h-4=0$
or $(1+m^2)h^2+2h(6m-2)+32=0$, it is a quadratic equation in... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Given $f(x+1/x) = x^2 +1/x^2$, find $f(x)$ Given $f(x+1/x) = x^2 +1/x^2$, find $f(x)$. Please show me the way you find it.
The answer in my textbook is $f(x)=\frac{1+x^2+x^4}{x\cdot \sqrt{1-x^2}}$
| Note that $(x+\frac1x)^2=x^2+2+\frac1{x^2}$.
Hence it looks like $f(x)=x^2-2$ is a good candidate.
Of course, $\left|x+\frac1x\right|\ge2$ implies that we cannot say anything about $f(x)$ if $|x|<2$.
But for $|x|\ge 2$, we can find a real number $t$ such that $t^2-xt+1=0$ (and hence $t+\frac1t=x$), namely $t=\frac{x\pm... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to relate $2\sin(3\pi/8)-2\sin(7\pi/8)$ and $\csc(3\pi/8)$? Trying to simplify $2\sin(3\pi/8)-2\sin(7\pi/8)$ down to $\csc(3\pi/8)$. The two expressions have equal decimal approximations but I'm literally at my wit's end trying to relate them based on trigonometric identities.
| $$\sin(7 \pi/8) = \sin(\pi/2 + 3 \pi/8) = \cos(3 \pi/8)$$
\begin{align}
\dfrac{\sin(3 \pi/8) - \sin(7 \pi/8)}{\csc(3 \pi/8)} & = \sin(3 \pi/8) (\sin(3 \pi/8) - \sin(7 \pi/8))\\
& = \sin(3 \pi/8) (\sin(3 \pi/8) - \cos(3 \pi/8))\\
& = \sin^2(3 \pi/8) - \sin(3 \pi/8) \cos(3 \pi/8)\\
& = \underbrace{\dfrac{1 - \cos(3 \pi/4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/222267",
"timestamp": "2023-03-29T00:00:00",
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Find all natural numbers with the property that... Find all natural numbers with the property that when the first digit is moved to the end, the resulting number is $3.5$ times the original number.
| Suppose that our original number is $k+1$ digits long. Let $a$ denote the first digit of the number. We represent the number as
$$10^ka + b$$
where $b < 10^k$ is the remaining portion of our number. The condition is thus expressed as
$$3.5(10^ka + b) = 10b + a$$
Clearing fractions and rearranging, we end up with
$$(7\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/223928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Positive integers and the number of their digits Let $a$, $b$, $c$ be positive integers and $s(a)$, $s(b)$, $s(c)$ denote the number of their digits (when the integers are written in decimal form) respectively. If,
$s(a)+s(b)=a\qquad$
$a + b + s(c) = c\qquad$
and
$4 + s(a) + s(b) + s(c) = b \qquad$
then what would b... | If $a$ is large, then (1) says that $b\approx10^a$; then (3) says that $c\approx10^{10^a}$, and then (2) can't work out. Thus $a$ can't be large. Then the same arguments applied to (2) and (3) show that $b$ also can't be large, and then the same argument applied just to (2) shows that $c$ can't be large, either. Thus t... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The positive integer solutions for $2^a+3^b=5^c$ What are the positive integer solutions to the equation
$$2^a + 3^b = 5^c$$
Of course $(1,\space 1, \space 1)$ is a solution.
| If $a=0$, then it is clear there are no solutions. If $b=0$, then we need $2^a + 1 = 5^c$. It is easy to show in this case $a=2,c=1$ is the only solution by showing that we need $2^{a-2}|c$. When $c=0$ there are obviously no solutions.
Suppose $a=1$. Then $2 + 3^b = 5^c$ only has the solution of $b=1,c=1$. To show this... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove inequality $\frac{a}{b^2 + 1} + \frac{b}{c^2 + 1} + \frac{c}{a^2 + 1} ≤ 1.5$ Prove $\frac{a}{b^2 + 1} + \frac{b}{c^2 + 1} + \frac{c}{a^2 + 1} ≥ 1.5$ with $a + b + c = 3 $ and $a,b,c > 0$
The correct question is $\frac{a}{b^2 + 1} + \frac{b}{c^2 + 1} + \frac{c}{a^2 + 1} ≥ 1.5$ (I have proved it)
Can prove $\fra... | The inequality does not hold! For $a = 2, b = c = 1/2$, we have
$$
\frac{a}{b^2 + 1} + \frac{b}{c^2 + 1} + \frac{c}{a^2 + 1} \geq \frac{a}{b^2 + 1} = \frac 8 5 > \frac 3 2
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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What do I use to find the image and kernel of a given matrix? I had a couple of questions about a matrix problem. What I'm given is:
Consider a linear transformation $T: \mathbb R^5 \to \mathbb R^4$ defined by $T( \vec{x} )=A\vec{x}$, where
$$A = \left(\begin{array}{crc}
1 & 2 & 2 & -5 & 6\\
-1 & -2 & -1 & 1 & ... | After a long night of studying I finally figured out the answer to these. The previous answers on transformation were all good, but I have the outlined steps on how to find $\mathrm{im}(T)$ and $\ker(T)$.
$$A = \left(\begin{array}{crc}
1 & 2 & 2 & -5 & 6\\
-1 & -2 & -1 & 1 & -1\\
4 & 8 & 5 & -8 & 9\\
3 & 6 &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/236541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "34",
"answer_count": 4,
"answer_id": 1
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Express the volume of an n-sphere in terms of the volume of an n-1 dimensional ball It's an exercise from Munkres "Analysis on Manifolds" Chapter 5, "Integrating a scalar function over a manifold".
Due to the suggestion, I'm repeating the question here:
Express the volume of an n-sphere in terms of the volume of an n-1... | Express the volume of $S^n(a)$ in terms of the volume
$B^{n-1}(a)$. [\it Hint: \rm Follow the pattern of Example 2.]
\paragraph{\bf{sln}.}
We can write
\begin{eqnarray}
S^n(a) &=& \{ (x_1, \cdots x_{n+1}) \; , \; x_1^2 + \cdots x_{n+1}^2 = a^2 \} \\
&=& \{ (x_1, \cdots x_{n-1}) \; , \; x_1^2 + \cdots x_{n-1}^2 = ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Positive integers less than $N$ not divisible by $4$ or $6$ How many positive integers less than $N$ are not divisible by $4$ or $6$ for some $N$?
| A naive and computational answer: look at congruence modulo 12.
*
*Empty (no natural numbers less than 1) = $0$
*${1}$ = $1$
*${1,2}$ = $2$
*${1, 2, 3}$ = $3$
*${1, 2, 3}$ = $3$
*${1, 2, 3, 5}$ = $4$
*${1, 2, 3, 5}$ = $4$
*${1, 2, 3, 5, 7}$ = $5$
*${1, 2, 3, 5, 7}$ = $5$
*${1, 2, 3, 5, 7, 9}$ = $6$
*${1, 2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/237650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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numbers' pattern It is known that
$$\begin{array}{ccc}1+2&=&3 \\ 4+5+6 &=& 7+8 \\
9+10+11+12 &=& 13+14+15 \\\
16+17+18+19+20 &=& 21+22+23+24 \\\
25+26+27+28+29+30 &=& 31+32+33+34+35 \\\ldots&=&\ldots
\end{array}$$
There is something similar for square numbers:
$$\begin{array}{ccc}3^2+4^2&=&5^2 \\ 10^2+11^2+12^2 &=& 13... | We do have
$$ \eqalign{6^3 &= 3^3 + 4^3 + 5^3\cr
20^3 &= 11^3 + 12^3 + 13^3 + 14^3\cr
40^3 &= 3^3 + \ldots + 22^3\cr
70^3 &= 15^3 + \ldots + 34^3\cr
37^3 + 38^3 &= 5^3 + \ldots + 25^3\cr
30^3 + 31^3 + 32^3 &= 7^3 + \ldots + 24^3\cr
101^3 + 102^3 + 103^3 &= 61^3 + \ldot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/238532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 0
} |
Find the sum of the first $n$ terms of $\sum^n_{k=1}k^3$ The question:
Find the sum of the first $n$ terms of $$\sum^n_{k=1}k^3$$
[Hint: consider $(k+1)^4-k^4$]
[Answer: $\frac{1}{4}n^2(n+1)^2$]
My solution:
$$\begin{align}
\sum^n_{k=1}k^3&=1^3+2^3+3^3+4^3+\cdots+(n-1)^3+n^3\\
&=\frac{n}{2}[\text{first term} + \text{la... | Method 1
Using the binomial identity
$$
\sum_{k=m}^{n-j}\binom{n-k}{j}\binom{k}{m}=\binom{n+1}{j+m+1}\tag{1}
$$
with $j=0$ yields
$$
\sum_{k=0}^n\binom{k}{m}=\binom{n+1}{m+1}\tag{2}
$$
Using $(2)$ and the identity
$$
k^3=6\binom{k}{3}+6\binom{k}{2}+\binom{k}{1}\tag{3}
$$
we get that
$$
\begin{align}
\sum_{k=0}^nk^3
&=6... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/239909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Evaluate $\int_1^4\frac{dx}{x^2+x+1}$ I've to evaluate the integral $$\int_1^4 \frac{dx}{x^2+x+1}$$
but I can't find the answer. I checked with Wolfram Alpha but I still don't fully understand. Could you please explain the steps to me? I think I should use arctan in my answer.
| First note that
$$\dfrac1{x^2 + x + 1} = \dfrac1{(x+1/2)^2 + (\sqrt{3}/2)^2}$$
We now have that
\begin{align}
I = \int_1^4 \dfrac{dx}{x^2 + x + 1} = \int_1^4 \dfrac{dx}{(x+1/2)^2 + (\sqrt{3}/2)^2}
\end{align}
Now let $y = x+1/2$, then we get that
$$I = \int_{3/2}^{9/2} \dfrac{dy}{y^2 + (\sqrt{3}/2)^2}$$
Now let $y = \s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/241676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
On Ceva's Theorem? The famous Ceva's Theorem on a triangle $\Delta \text{ABC}$
$$\frac{AJ}{JB} \cdot \frac{BI}{IC} \cdot \frac{CK}{EK} = 1$$
is usually proven using the property that the area of a triangle of a given height is proportional to its base.
Is there any other proof of this theorem (using a different... | The following proof is heavily influenced by my background in projective geometry. Use homogenous and choose the following affine basis, without loss of generality. You might also consider this as barycentric coordinates, the way Olivier Bégassat wrote in a comment.
\begin{align*}
A &= \begin{pmatrix}1\\0\\0\end{pmatr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 1
} |
Limits of functions with square root and sin Can someone help me calculate the following limits(without L'Hopital!) :
1) $\lim_{x\to 1 } \frac { \sqrt{x}-1}{\sqrt[3]{x}-1} $ . I have tried taking the logarithm of this limit, but without any success.
2) $\lim_{x\to\pi} \frac{\sin5x}{\sin3x}$ and the hint is : $\sin(\p... | For the first one, set $\sqrt[6]{x} = y$. This means $\sqrt{x} = y^3$ and $\sqrt[3]{x} = y^2$. We then get that
$$\lim_{x \to 1} \dfrac{\sqrt{x}-1}{\sqrt[3]{x}-1} = \lim_{y \to 1} \dfrac{y^3-1}{y^2-1} = \lim_{y \to 1} \dfrac{(y-1)(y^2+y+1)}{(y-1)(y+1)} = \lim_{y \to 1} \dfrac{(y^2+y+1)}{(y+1)} = \dfrac32$$
For the seco... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Value of $\lim_{n\to \infty}\frac{1^n+2^n+\cdots+(n-1)^n}{n^n}$ I remember that a couple of years ago a friend showed me and some other people the following expression:
$$\lim_{n\to \infty}\frac{1^n+2^n+\cdots+(n-1)^n}{n^n}.$$
As shown below, I can prove that this limit exists by the monotone convergence theorem. I als... | We can write $a_n=\sum_{k=1}^{n-1}\left(1-\frac{k}{n}\right)^n$. The given limit can then be written in the form
$$
\lim_{n\to\infty}\sum_{k=1}^\infty\left[1-\frac{k}{n}\right]_+^n
$$
(where $[x]_+$ is the positive part, i.e., $[x]_+=x$ if $x\geq0$, $0$ otherwise).
I claim that for fixed $k$, $\left[1-\frac{k}{n}\right... | {
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"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 0
} |
Example of two dependent random variables that satisfy $E[f(X)f(Y)]=Ef(X)Ef(Y)$ for every $f$ Does anyone have an example of two dependent random variables, that satisfy this relation?
$E[f(X)f(Y)]=E[f(X)]E[f(Y)]$
for every function $f(t)$.
Thanks.
*edit: I still couldn't find an example. I think one should be of two i... | Here is a counterexample. Let $V$ be the set $\lbrace 1,2,3 \rbrace$. Consider random variables $X$ and $Y$ with values in $V$, whose joint distribution is defined by the following matrix :
$$
P=\left(
\begin{matrix}
\frac{1}{10} & 0 & \frac{1}{5} \\
\frac{1}{5} & \frac{1}{10} & 0 \\
\frac{1}{30} & \frac{7}{30} & ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/245050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 5,
"answer_id": 1
} |
Partial Differential with Integrating Factor I have the following:
$ \dfrac{dx}{dz} = \dfrac{x(x+y)}{(y-x)(2x+2y+z(x,y))} $
I've tried to solve through using an integrating factor, i.e. I rearranged to get:
$ \dfrac{dz(x,y)}{dx} - \dfrac{(y-x)}{x(x+y)}z(x,y) = \dfrac{2(y-x)}{x} $
and used IF $= \dfrac{(x+y)^2}{x} $. I ... | It would be better in this context either to use partial derivative sign, or suppress $y$ as an argument and treat it purely as a parameter. To cross-check your solution a calculation by means of variating the constant of integration can be carried out.
First solve the homogeneous equation
$$\frac{dz}{z}=\frac{y-x}{x\l... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Proof using Mod How can you prove that:
$$a^7\equiv a\:(\text{mod } 42)$$
I haven't been given any other information other than to use Fermat's theorem.
| There are two versions of Fermat's Theorem.
Version $1$: If $a$ is not divisible by $p$, then $a^{p-1}\equiv 1\pmod{p}$.
Version $2$: For any $a$, $a^p\equiv a\pmod p$.
The more common version is Version $1$. From it, you can easily deduce Version $2$. For if $a$ is not divisible by $p$, then $a^{p-1}\equiv 1\pmod{p}$... | {
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"url": "https://math.stackexchange.com/questions/245378",
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"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
Proof : $2^{n-1}\mid n!$ if and only if $n$ is a power of $2$. I want to prove that:
$2^{n-1}\mid n!$ if and only if $n$ is a power of $2$.
| write out n! and consider how many times each divides by 2 for example 16:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 = 16!
^ ^ ^ ^ ^ ^ ^ ^ divisible by 2
^ ^ ^ ^ divisible by 2^2
^ ^ divisible by 2^3
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/248082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Put into each cell one number 1 or one number -1 that the sums of number in each coloums each rows is 0. There is a 4 x 4 grid. In each cell of the grid, you are allowed to put either a 1 or a -1. The sum of the numbers of each column and each row must equal 0. How many such configurations of the grid are there?
| You can to start completing the first line of the square
\begin{array}{|c|c|c|c|}
\hline
.& . & .&. \\ \hline
.& .&. & .\\ \hline
. & . & .& .\\ \hline
.& . & .& .\\ \hline
\end{array}
And you can do this of $ 6 = \binom{4}{2} $ ways because you need to choose the 2 places between the 4 to put the numbers 1. For ex... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Closed form for $\sum_{k=0}^{n} \binom{n}{k}\frac{(-1)^k}{(k+1)^2}$ How can I calculate the following sum involving binomial terms:
$$\sum_{k=0}^{n} \binom{n}{k}\frac{(-1)^k}{(k+1)^2}$$
Where the value of n can get very big (thus calculating the binomial coefficients is not feasible).
Is there a closed form for this su... | I'm even later to the party, but that's only because "absorption identity" kept yelling in my ear. :)
One application of the absorption identity gets one of the factors of $k+1$ out of the denominator:
$$\begin{align}
\sum_{k=0}^n \binom{n}{k} \frac{(-1)^k}{(k+1)^2} &= \frac{1}{n+1} \sum_{k=0}^n \binom{n+1}{k+1} \frac{... | {
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"url": "https://math.stackexchange.com/questions/254416",
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"source": "stackexchange",
"question_score": "16",
"answer_count": 9,
"answer_id": 0
} |
How to show this is relatively prime? Let $a_1= 2$, and for each
$y > 1$, define $a_{y+1} = a_y(a_y −1) +1$.
Prove that for all $x \ne y$, $a_x$ and $a_y$ are coprime.
| Hint:
$a_{y+1}=a_y(a_y-1)+1$
$\implies a_{y+1}-1=a_y(a_y-1) $
So, $ a_y-1=a_{y-1}(a_{y-1}-1)$ and $a_{y+1}-1=a_ya_{y-1}(a_{y-1}-1) $
$ a_{y-1}-1=a_{y-2}(a_{y-2}-1)$ and
so $a_{y+1}-1=a_ya_{y-1}a_{y-2}(a_{y-2}-1)=(a_1-1)\prod_{y\le x \le1}a_x=\prod_{y\le x \le1}a_x$ as $a_1=2$
As $(a_{y+1}-1,a_{y+1})=1, (\prod_{y\le x... | {
"language": "en",
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"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Evaluate $\lim\limits_{x \to \infty}\left (\sqrt{\frac{x^3}{x-1}}-x\right)$ Evaluate
$$
\lim_{x \to \infty}\left (\sqrt{\frac{x^3}{x-1}}-x\right)
$$
The answer is $\frac{1}{2}$, have no idea how to arrive at that.
| Using $$\alpha-\beta=\frac{\alpha^2-\beta^2}{\alpha+\beta}$$
yields
$$\lim_{x \to \infty} \sqrt{\frac{x^3}{x-1}}-x=\lim_{x \to \infty} \frac{\frac{x^3}{x-1}-x^2}{\sqrt{\frac{x^3}{x-1}}+x}=\lim_{x \to \infty} \frac{x^3-x^3+x^2}{(x-1)(\sqrt{\frac{x^3}{x-1}}+x)}=
\lim_{x \to \infty} \frac{x^2}{\sqrt{x^4-x^3}+x^2-x}=\lim_{... | {
"language": "en",
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"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 0
} |
How to integrate this $\int\frac{\mathrm{d}x}{{(4+x^2)}^{3/2}} $ without trigonometric substitution? I have been looking for a possible solution but they are with trigonometric integration..
I need a solution for this function without trigonometric integration
$$\int\frac{\mathrm{d}x}{{(4+x^2)}^{3/2}}$$
| Let $I=\int \frac{dx}{\sqrt{x^2+4}} \,;\, J=\int \frac{dx}{\sqrt{x^2+4}^3}$
We integrate $I$ by parts, we get:
$u=(x^2+4)^{-1/2}, dv=dx$
$du=-x(x^2+4)^{-3/2}, v=x$
Thus
$$I= \frac{x}{\sqrt{x^2+4}}+ \int \frac{x^2}{\sqrt{x^2+4}^3}= \frac{x}{\sqrt{x^2+4}}+ \int \frac{x^2+4-4}{\sqrt{x^2+4}^3}=$$
$$= \frac{x}{\sqrt{x^2+4}}... | {
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"source": "stackexchange",
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"answer_count": 4,
"answer_id": 0
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Using product and chain rule to find derivative. Find the derivative of
$$y =(1+x^2)^4 (2-x^3)^5$$
To solve this I used the product rule and the chain rule.
$$u = (1+x^2)^4$$
$$u' = 4 (1+x^2)^3(2x)$$
$$v= (2-x^3)^5$$
$$v' = 5(2-x^3)^4(3x^2)$$
$$uv'+vu'$$
$$((1+x^2)^4)(5(2-x^3)^4(3x^2)) + ((2-x^3)^5 )(4 (1+x^2)^3(2x... | Everything is correct in your answer, except for the chain rule for $v$.
The derivative of $2-x^3$ is $-3x^2$.
So $v'=5(2-x^3)^4(-3x^2)$ and this is why the $15x^2$ becomes negative.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/261011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Help Solving Trigonometry Equation I am having difficulties solving the following equation:
$$4\cos^2x=5-4\sin x$$
Hints on how to solve this equation would be helpful.
|
I am having difficulties solving the following equation:$${4\cos^2x=5-4\sin x}$$
First, substitute $4\cos^2(x)$ with $4\left(1 - \sin^2(x)\right) = 4 - 4\sin^2(x).$ We are left with $4 - 4\sin^2(x) = 5 - 4 \sin (x) .$ This can be rewritten as $-4\sin^2(x) + 4\sin(x) - 1 = 0.$
Observe that the equation can further b... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
What is the value of $N$? $N$ is the smallest positive integer such that for any integer $n > N$, the quantity $n^3 – 7n^2 + 11n – 5$ is positive. What is the value of $N$?
Note: $N$ is a single digit number.
| Start by trying to factorise the expression. One thing to note is that the sum of the coefficients is zero: $(+1) + (-7) + (+11) + (-5) = 0$, and so $n=1$ is a solution. We are able to factorise:
$$ n^3 - 7n^2 + 11n -5 \equiv (n-1)(\text{quadratic}) \, . $$
Assume the quadratic is of the form $a_2n^2+a_1n+a_0$, for som... | {
"language": "en",
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"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
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Prove that $\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+a}\geq\frac{1}{2}(a+b+c)$ for positive $a,b,c$ Prove the following inequality: for
$a,b,c>0$
$$\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+a}\geq\frac{1}{2}(a+b+c)$$
What I tried is using substitution:
$p=a+b+c$
$q=ab+bc+ca$
$r=abc$
But I cannot reduce $a^2(b+c)... | This is just Cauchy-Schwarz:
$$\left(\frac{a}{\sqrt{a+b}}\sqrt{a+b}+\frac{b}{\sqrt{b+c}}\sqrt{b+c}+\frac{c}{\sqrt{a+c}}\sqrt{a+c}\right)^2 \leq \left(\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+a} \right)\big( (a+b)+(a+c)+(b+c)\big)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/264931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 1
} |
Show that the sum of the largest odd divisors of $n+1, n+2, \ldots, 2n$ (where $n$ is a natural number) is a perfect square? I've been given the solution but I don't understand it at all, could someone please explain?
| The first thing to understand is that every positive integer $N$ can be written uniquely in the form $2^ab$, where $a$ is a non-negative integer, and $b$ is a positive odd integer. For example, $12=2^2\cdot3$ ($a=2,b=3$), $11=2^0\cdot 11$ ($a=0,b=11$), and $8=2^3\cdot1$ ($a=3,b=1$). In this decomposition the number $b$... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Origin of the constant $\phi$ in Binet's formula of the $n$-th term of the Fibonacci sequence I have read in a page that " To find the nth term of the Fibonacci series, we can use Binet's Formula"
F(n) = round( (Phi ^ n) / √5 ) provided n ≥ 0
where Phi=1·61803 39887 49894 84820 45868 34365 63811 77203 09179 805... | Using generating functions:
\begin{align}
\color{brown}{f(x) = \sum_0^\infty F_n x^n} &= F_0x^0 + F_1x^1+F_2 x^2 + F_3 x^3+ F_4 x^4+\cdots\\
xf(x) &= \quad\quad\quad\, F_0x^1+F_1x^2+F_2x^3+F_3x^4+\cdots\\
x^2f(x)&=\quad\quad\quad\quad\quad\quad\,\,F_0x^2+F_1x^3 + F_2x^4+\cdots
\end{align}
Since the value of $F_0=0$ a... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
how many number like $119$ How many 3-digits number has this property like $119$:
$119$ divided by $2$ the remainder is $1$
119 divided by $3$ the remainderis $2 $
$119$ divided by $4$ the remainder is $3$
$119$ divided by $5$ the remainder is $4$
$119$ divided by $6$ the remainder is $5$
| Note that
$$x \equiv 1 \mod 2 \\ x \equiv 2 \mod 3 \\ x \equiv 3 \mod 4 \\ x \equiv 4 \mod 5 \\ x \equiv 5 \mod 6$$
is equivalent to
$$x \equiv -1 \mod 2 \\ x \equiv -1 \mod 3 \\ x \equiv -1 \mod 4 \\ x \equiv -1 \mod 5 \\ x \equiv -1 \mod 6$$
And by the Chinese Remainder Theorem, the solution to the latter is easily f... | {
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"url": "https://math.stackexchange.com/questions/268619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 0
} |
Differential equation $xy^3y'=2y^4+x^4$ Solve the differential equation
$$xy^3y'=2y^4+x^4$$
| Another method is to realise that $y^3y'$ is, up to a constant factor, the derivative of $y^4$. So we obtain:
\begin{align*}
xy^3y' &= 2y^4 + x^4\\
x\frac{d}{dx}\left(\frac{y^4}{4}\right) - 2y^4 &= x^4\\
\frac{d}{dx}(y^4) - \frac{8}{x}y^4 &= 4x^3\\
\end{align*}
where, in the last step, we need $x \neq 0$. Now you can s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/269086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 2
} |
$ \displaystyle\lim_{n\to\infty}\frac{1}{\sqrt{n^3+1}}+\frac{2}{\sqrt{n^3+2}}+\cdots+\frac{n}{\sqrt{n^3+n}}$ $$ \ X_n=\frac{1}{\sqrt{n^3+1}}+\frac{2}{\sqrt{n^3+2}}+\cdots+\frac{n}{\sqrt{n^3+n}}$$ Find $\displaystyle\lim_{n\to\infty} X_n$ using the squeeze theorem
I tried this approach:
$$
\frac{1}{\sqrt{n^3+1}}\le\frac... | Since
$$
X_n=\sum_{k=1}^n\frac{k}{\sqrt{n^3+k}}\ge\sum_{k=1}^n\frac{k}{\sqrt{n^3+n}}=\frac{n^2+n}{2\sqrt{n^3+n}},
$$
it follows that
$$
\lim_{n \to \infty}X_n \ge \lim_{n \to \infty}\frac{n^2+n}{2\sqrt{n^3+n}}=\infty,
$$
i.e. $\lim_{n \to \infty}X_n=\infty$.
| {
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"url": "https://math.stackexchange.com/questions/270216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Recurrence relation by substitution I have an exercise where I need to prove by using the substitution method the following
$$T(n) = 4T(n/3)+n = \Theta(n^{\log_3 4})$$
using as guess like the one below will fail, I cannot see why, though, even if I developed the substitution
$$T(n) ≤ cn^{\log_3 4}$$
finally, they ask m... | Let $n = 3^{m+1}$. Then we have
$$T \left(3^{m+1} \right) = 4T \left(3^{m} \right) + 3^{m+1}$$
Let $T\left(3^m\right) = g(m)$. We then get that
$$g(m+1) = 4g(m) + 3^{m+1} = 4(4g(m-1) + 3^m) + 3^{m+1} = 4^2 g(m-1) + 4\cdot 3^m + 3^{m+1}\\
= 4^2 (4g(m-2) + 3^{m-1}) + 4\cdot 3^m + 3^{m+1} = 4^3 g(m-3) + 4^2 \cdot 3^{m-1} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/270591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Find basis of im, ker and dim im, dim ker verification In my homework, I've to find basis of im and ker of linear transformation
$\varphi:\mathbb{R}^{3}\rightarrow\mathbb{R}^{2}, \varphi((x_{1},x_{2},x_{3}))=(2x_{1}+x_{2}-3x_{3},x_{1}+4x_{2}+2x_{3})$
My solution:
Kernel of $\varphi$ is described by a matrix representin... | Kernel of $\varphi$ is described by a matrix representing set of equations
$\left[ \begin{matrix}2 & 1 & -3 & 0\\ 1 & 4 & 2 & 0 \end{matrix}\right] \sim\left[ \begin{matrix}0 & -7 & -7 & 0\\ 1 & 4 & 2 & 0 \end{matrix}\right] \sim\left[ \begin{matrix}0 & 1 & 1 & 0\\ 1 & 4 & 2 & 0 \end{matrix}\right]$
General solution ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/272762",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Need help to solve the trigonometric equation $\tan\frac {1+x}{2} \tan \frac {1+x}{3}=-1$ How can I solve the equation $\tan\frac {1+x}{2} \tan \frac {1+x}{3}=-1$. Please give me some hint for that. Thank you.
| Assuming $\frac12,\frac13$ are measured in radian,
$$\tan\left(\frac{1+ x}2\right)=-\frac1{\tan\left(\frac{1+ x}3\right)}=-\cot\left(\frac{1+ x}3\right)=\tan\left(\frac\pi2+\frac{x+1}3\right)$$ as $\tan(\frac\pi2+C)=-\cot C$
So, $$\frac{1+ x}2=n\pi+\frac\pi2+\frac{x+1}3$$ where $n$ is any integer as $\tan A=\tan B\impl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/273359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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I am trying to solve the inequality $\log_{\log{\sqrt{9-x^2}}} x^2 <0$ I am trying to solve the inequality
$$\log_{\log{\sqrt{9-x^2}}} x^2 <0.$$
I got $\mathrm{S.S}=(-\sqrt8 ,-1)\cup( 1,\sqrt8)$, but a friend got $\mathrm{S.S}=(-1,1)- \{0\}$.
Please, what is true?
| We shall consider two cases:
(1) Case ($\log \sqrt {9-x^2} >1$)
$$\log_{\log \sqrt{9-x^2}} x^2 <0 \Rightarrow $$
$$\left \{
\begin{array}{l}
0 \neq x^2 < 1 \\
\log \sqrt {9-x^2} >1 \\
\end{array}
\right. \Leftrightarrow
\left \{
\begin{array}{l}
-1 < x < 1 \quad \mathrm {and} \quad x\neq 0\\
\sqrt {9-x^2} > 10 \\
\end{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/275727",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Finding polynomial function's zero values not native English speaker so I may get some terms wrong and so on.
On to the question:
I have as an assignment to find a polynomial function $f(x)$ with the coefficients $a$, $b$ and $c$ (which are all integers) which has one root at $x = \sqrt{a} + \sqrt{b} + \sqrt{c}$.
I've ... | You certainly have the right idea. Following Gerry Myerson's suggestion above, we have
\begin{align}
x&=\sqrt{a}+\sqrt{b}+\sqrt{c}\\
x-\sqrt{a}&=\sqrt{b}+\sqrt{c}\\
(x-\sqrt{a})^2=x^2-2x\sqrt{a}+a&=b+2\sqrt{bc}+c=(\sqrt{b}+\sqrt{c})^2\\
x^2+a-b-c&=2x\sqrt{a}+2\sqrt{bc}\\
\left(x^2+a-b-c\right)^2&=4x^2a+8x\sqrt{abc}+4bc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/276111",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Is my limit evaluation correct? I'm studying for my calculus exam and I have the following limit:
$$\lim\limits_{n \to \infty} \left ( \frac{1}{\sqrt{n^3 +3}}+\frac{1}{\sqrt{n^3 +6}}+ \cdots +\frac{1}{\sqrt{n^3 +3n}} \right )$$
My solution is:
$$\begin{align*}
&\lim\limits_{n\ \to \infty} \left ( \frac{1}{\sqrt{n^3 +3}... | The answer is correct, the reasoning is not. First we solve the problem.
The sum has $n$ terms. Each term is $\lt \frac{1}{n^{3/2}}$. It follows that if $S_n$ is our sum, then
$$0\lt S_n \lt \frac{n}{n^{3/2}}=\frac{1}{n^{1/2}}.$$
Now let $n\to\infty$. Note that $\frac{1}{n^{1/2}}\to 0$, so by Squeezing $S_n\to 0$.
Re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/276178",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
How to show that $\frac{x^2}{x-1}$ simplifies to $x + \frac{1}{x-1} +1$ How does $\frac{x^2}{(x-1)}$ simplify to $x + \frac{1}{x-1} +1$?
The second expression would be much easier to work with, but I cant figure out how to get there.
Thanks
| A general $^1$ method is to perform the polynomial long division algorithm in the following form or another equivalent one:
to get
$$
x^{2}=(x-1)(x+1)+1.
$$
Divide both sides by $x-1$:
$$\begin{eqnarray*}
\frac{x^{2}}{x-1} &=&\frac{(x-1)(x+1)+1}{x-1}=\frac{(x-1)(x+1)}{x-1}+\frac{1}{x-1} \\
&=&\frac{x+1}{1}+\frac{1}{x-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/278481",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 1
} |
Some weird equations In our theoreticall class professor stated that from this equation $(C = constant)$
$$
x^2 + 4Cx - 2Cy = 0
$$
we can first get:
$$
x = \frac{-4C + \sqrt{16 C^2 - 4(-2Cy)}}{2}
$$
and than this one:
$$
x = 2C \left[\sqrt{1 + \frac{y}{2C}} -1\right]
$$
How is this even possible?
| This comes from the quadratic formula. See here for more details.
In your case, $$x^2 + 4Cx -2Cy = x^2 + 4Cx +(2C)^2 - (2C)^2-2Cy = (x+2C)^2 - (2C)^2 \left(1 + \dfrac{y}{2C} \right) = 0$$
Hence, we have that
$$(x+2C)^2 = (2C)^2 \left(1 + \dfrac{y}{2C} \right) \implies x + 2C = \pm 2C \sqrt{\left(1 + \dfrac{y}{2C} \righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/278780",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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Isolate middle value from $3\times 3$ matrix Not trained in math the solution to this problem is not immediately apparent, plus I am working on a larger problem which I'd rather get to.
I'm trying to isolate the middle value from a $3\times 3$ matrix
Suppose my matrix is
$$\begin{pmatrix}
\\ 5 & 7 & 3
\\ 4 & 13 & 9
\\... | $$\begin{pmatrix}0&0&0\\0&1&0\\0&0&0\end{pmatrix}\begin{pmatrix}5&7&3\\4&13&9\\9&9&1\end{pmatrix}\begin{pmatrix}0&0&0\\0&1&0\\0&0&0\end{pmatrix} = \begin{pmatrix}0&0&0\\0&13&0\\0&0&0\end{pmatrix} $$
The first matrix isolates the middle row and the last isolates the middle column.
Edit: These types of matrices are somet... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/280274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Studying $ u_n = \int_0^1 (\arctan x)^n \mathrm dx$ I would like to find an equivalent of:
$$ u_n = \int_0^1 (\arctan x)^n \mathrm dx$$
which might be: $$ u_n \sim \frac{\pi}{2n} \left(\frac{\pi}{4} \right)^n$$
$$ 0\le u_n\le \left( \frac{\pi}{4} \right)^n$$
So $$ u_n \rightarrow 0$$
In order to get rid of $\arctan x$... | $$
\begin{align}
&\int_0^1\arctan^n(x)\,\mathrm{d}x\\
&=\int_0^{\pi/4}x^n\sec^2(x)\,\mathrm{d}x\\
&=\int_0^{\pi/4}\left(\frac\pi4-x\right)^n\sec^2\left(\frac\pi4-x\right)\,\mathrm{d}x\\
&=\frac{(\pi/4)^{n+1}}{n}\int_0^ne^{-x}\color{#C00000}{e^x\left(1-\frac{x}{n}\right)^n}\color{#00A000}{\sec^2\left(\frac\pi4-\frac{\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/281017",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Finding $\frac{1}{S_1}+\frac{1}{S_2}+\frac{1}{S_3}+\cdots+\frac{1}{S_{2013}}$ Assume $S_1=1 ,S_2=1+2,S=1+2+3+,\ldots,S_n=1+2+3+\cdots+n$
How to find :
$$\frac{1}{S_1}+\frac{1}{S_2}+\frac{1}{S_3}+\cdots+\frac{1}{S_{2013}}$$
| $$\sum_{n=1}^{2013}\frac{1}{n(n+1)/2}=\sum_{n=1}^{2013}\frac{2}{n(n+1)}=2\sum_{n=1}^{2013}\frac{1}{n(n+1)}=2\sum_{n=1}^{2013}\left(\frac{1}{n}-\frac{1}{n+1}\right)=$$
$$2\sum_{n=1}^{2013}\frac{1}{n}-2\sum_{n=1}^{2013}\frac{1}{n+1}=2\sum_{n=1}^{2013}\frac{1}{n}-2\sum_{n=2}^{2014}\frac{1}{n}=2\sum_{n=1}^{2013}\frac{1}{n}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/281318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Let $a,b$ and $c$ be real numbers.evaluate the following determinant: |$b^2c^2 ,bc, b+c;c^2a^2,ca,c+a;a^2b^2,ab,a+b$| Let $a,b$ and $c$ be real numbers. Evaluate the following determinant:
$$\begin{vmatrix}b^2c^2 &bc& b+c\cr c^2a^2&ca&c+a\cr a^2b^2&ab&a+b\cr\end{vmatrix}$$
after long calculation I get that the answer w... | We can assume $\,abc\neq 0\,$ , otherwise the determinant is zero at once (why? For example, suppose $\,b=0\,$ . Then either $\,ac=0\,$ and we get a row of zeros, or the 3rd row is a multiple of the 1st one...and likewise if $\,a=0\vee c=0\,$ ):
$$\begin{vmatrix}b^2c^2 &bc& b+c\cr c^2a^2&ca&c+a\cr a^2b^2&ab&a+b\cr\end{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/282655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 2
} |
Derivative for log I have the following problem:
$$
\log \bigg( \frac{x+3}{4-x} \bigg)
$$
I need to graph the following function so I will need a starting point, roots, zeros, stationary points, inflection points and local minimum and maximum and I need to know where the function grows and declines.
I calculated roo... | For the stationary points you need to find $\frac{d}{dx} \left(\log \bigg( \frac{x+3}{4-x} \bigg)\right)$. You need to use the chain rule here $\frac{d}{dx}\log(f(x))=\frac{1}{f(x)}\cdot f'(x)$ which would give:
$$\frac{d}{dx} \left(\log \bigg( \frac{x+3}{4-x} \bigg)\right)= \frac{1}{\frac{x+3}{4-x}} \cdot \frac{d}{dx}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/284193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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How to find the solution of the differential equation
Find the solution of the differential equation
$$\frac{dy}{dx}=-\frac{x(x^2+y^2-10)}{y(x^2+y^2+5)}, y(0)=1$$
Trial: $$\begin{align} \frac{dy}{dx}=-\frac{x(x^2+y^2-10)}{y(x^2+y^2+5)} \\ \implies \frac{dy}{dx}=-\frac{1+(y/x)^2-10/x^2}{(y/x)(1+(y/x)^2+5/x^2)} \\ ... | Your differential equation can be written as
$$ (x^{3}+x y^{2}-10 x)dx+(x^{2}y+y^{3}+5y)dy=0$$
So it is of the form $M(x,y)dx+N(x,y)dy=0$
Now, $$M_{y}=2xy$$ and $$N_{x}=2xy.$$
$\therefore$ The given differential equation is exact.
So its general solution is
$$\int M dx+\int (\text{Terms in $N$ which does not contain $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/285114",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Consider a function $f(x) = x^4+x^3+x^2+x+1$, where x is an integer, $x\gt 1$. What will be the remainder when $f(x^5)$ is divided by $f(x)$? Consider a function $f(x) = x^4+x^3+x^2+x+1$, where x is an integer, $x\gt 1$. What will be the remainder when $f(x^5)$ is divided by $f(x)$ ?
$f(x)=x^4+x^3+x^2+x+1$
$f(x^5)=x^{2... | Rewrite the expression as
$$(x^{20}-1) +(x^{15}-1)+(x^{10}-1)+(x^5-1)+1+4.$$
The expression $x^4+x^3+x^2+x+1$ divides the first four terms, since $x^5-1$ does. So the remainder is $5$.
The idea obviously generalizes.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/285594",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
How to show $AB^{-1}A=A$
Let $$A^{n \times n}=\begin{pmatrix} a & b &b & \dots & b \\ b & a &b & \dots & b \\ b & b & a & \dots & b \\ \vdots & \vdots & \vdots & & \vdots \\ b & b & b & \dots &a\end{pmatrix}$$
where $a \neq b$ and $a + (n - 1)b = 0$.
Suppose $B=A+\frac{11'}{n}$ , where $1=(1,1,\dots,1)'$ is an $n ... | Writing replacing $a$ with $(1-n)b$ and $\mathbf{1}\mathbf{1}^T$ with $E$, we have
$$A = b\left( -n I + E \right) \qquad B = - n b I + \frac{1+nb}{n} E$$
Note that $E^2=nE$, so that
$$A^2 = b^2 \left(n^2I-2nE+E^2\right) = -nbA \qquad \text{and}\qquad AE=0$$
By the ever-popular Sherman-Morrison identity,
$$B^{-1} = -\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/287917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Counting Card hands with various restrictions I would like to know if my solutions are correct for the following three combinatorial card questions. In each question, assume we have a standard deck of cards (13 ranks, and 4 suits).
*
*How many ways are there to select $7$ cards so that we have at least $2$ distinct... | For the first question, there are $4\dbinom{13}{7}$ "bad" (one suit) hands, so the answer needs to be changed somewhat.
For the second question, there are $\dbinom{13}{7}$ ways to select the ranks that will be represented. For each rank, there are $4$ ways to choose the actual card, for a total of $\dbinom{13}{7}4^7$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/288538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A functional equation problem on $\mathbb{Q}^{+}$: $f(x)+f\left(\frac{1}{x}\right)=1$ and $f(2x)=2f\bigl(f(x)\bigr)$
Let $f$ be a function which maps $\mathbb{Q}^{+}$ to $\mathbb{Q}^{+}$ and satisfies
$$
\begin{cases}
f(x)+f\left(\frac{1}{x}\right)=1\\
f(2x)=2f\bigl(f(x)\bigr)
\end{cases}
$$
Show that $f\left(\frac{20... | Use induction on $q$. There are gaps in the proof which I cannot fill yet.
First establish the hypothesis for $q=1$.
We already know that $f(1)=1$ and $f(2)=2/3$. Let's assume that $f(x)=\frac{x}{x+1}$ for all integer $x<=2n$. Then we have: $
f(\frac{1}{2n})=\frac{1}{2n+1}; $ hence $2f(f(\frac{1}{2n}))=2f(\frac{1}{2n+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/289626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
} |
The figure below shows the graph of a function $f(x)$. How many solutions does the equation $f(f(x)) = 15$ have? From the graph we can find that, $f(-5)=f(-1)=f(9)=0$
| We are to find the solutions for $f(f(x)) = 15$.
From the graph, $f(4) = 15$ and $f(12) = 15$.
The required solutions will be those values of $x$ for which $f(x) = 4$ and $f(x) = 12$.
From the graph, the value of function $f(x) = 4$ at four different values of $x$, namely $x = –8, 1, 7.5, 10$.
The value of the function... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/292119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Computing the value of $\operatorname{Li}_{3}\left(\frac{1}{2} \right) $
How to prove the following identity
$$ \operatorname{Li}_{3}\left(\frac{1}{2} \right) = \sum_{n=1}^{\infty}\frac{1}{2^n n^3}= \frac{1}{24} \left( 21\zeta(3)+4\ln^3 (2)-2\pi^2 \ln2\right)\,?$$
Where $\operatorname{Li}_3 (x)$ is the trilogarithm... | Here is a partial solution:
Substitute $x=2\sin^2 \theta$ to obtain:
$$\int_0^{\frac{\pi}{4}} \left(\ln^2 2 + 4 \ln 2 \ln (\sin \theta) + 4 \ln^2 (\sin \theta)\right) \tan \theta \, \text{d}\theta$$
The $\ln^2 2$ part is trivial and evaluates to $\frac{\ln^3 2}{2}$.
The $\ln 2 \ln (\sin \theta)$ part is relatively stra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/293724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 2
} |
Factoring $3x^2 - 10x + 5$ How can $3x^2 - 10x + 5$ be factored? FOIL seemingly doesn't work (15 has no factors that sum to -10).
| You can always use the general formula for a quadratic equation, if we have $ax^2+bx+c$ then let $$x_1=\frac{-b+\sqrt{b^2-4ac}}{2a}$$ and $$x_2=\frac{-b-\sqrt{b^2-4ac}}{2a}$$ then we can write $$ax^2+bx+c=a(x-x_1)(x-x_2)$$ so in particular if we have $3x^2-10x+5$ then $$x_1=\frac{10+\sqrt{40}}{6}$$ and $$x_2=\frac{10-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/297667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 0
} |
Evaluating $\sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + \ldots}}}}$ Inspired by Ramanujan's problem and solution of $\sqrt{1 + 2\sqrt{1 + 3\sqrt{1 + \ldots}}}$, I decided to attempt evaluating the infinite radical
$$
\sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + \ldots}}}}
$$
Taking a cue from Ramanujan's solution method, I defi... | I'd rather try this:
If
$$x=\sqrt{ 1+\sqrt{2+\sqrt{4+\sqrt{8+\ldots}}}},$$
then
$$\begin{align}ux&=u\sqrt{ 1+\sqrt{2+\sqrt{4+\sqrt{8+\ldots}}}}\\
&=\sqrt{ u^2+u^2\sqrt{2+\sqrt{4+\sqrt{8+\ldots}}}}\\
&=\sqrt{ u^2+\sqrt{2u^4+u^4\sqrt{4+\sqrt{8+\ldots}}}}\\
&=\sqrt{ u^2+\sqrt{2u^4+\sqrt{4u^8+u^8\sqrt{8+\ldots}}}}\\
&=\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/300299",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 6,
"answer_id": 5
} |
inverse of function Thanks for the help! Here is the solution..
i had a problem:
$$f(x)=\frac{(\sqrt x+8)}{(5-\sqrt x)}$$
i had to find the inverse, so lets begin...
1) first i write in terms of $y$
$$y=\frac{(\sqrt x+8)}{(5-\sqrt x)}$$
2) now try to get $x$ by itself
$$(5-\sqrt x)y=\sqrt x+8$$
3)Distribute y along t... | I have a problem:
$$f(x)= \frac{\sqrt{x}+8}{5-\sqrt{x}}$$
I have to find the inverse but my calculations are off. I have listed what I did below, could someone tell me where I went wrong? Thank You in advance.
1) Write in terms of y
$$y = \frac{\sqrt{x}+8}{5-\sqrt{x}}$$
2) Now try to get $x$ by itself
$$ y(5-\sqrt{x}) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/300739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve $\frac{(b-c)(1+a^2)}{x+a^2}+\frac{(c-a)(1+b^2)}{x+b^2}+\frac{(a-b)(1+c^2)}{x+c^2}=0$ for $x$ Is there any smart way to solve the equation: $$\frac{(b-c)(1+a^2)}{x+a^2}+\frac{(c-a)(1+b^2)}{x+b^2}+\frac{(a-b)(1+c^2)}{x+c^2}=0$$
Use Maple I can find $x \in \{1;ab+bc+ca\}$
| $$\frac{(b-c)(1+a^2)}{x+a^2}+\frac{(c-a)(1+b^2)}{x+b^2}+\frac{(a-b)(1+c^2)}{x+c^2}=0$$
$$\Leftrightarrow (b-c)(1+a^2)+\frac{(x+a^2)(c-a)(1+b^2)}{x+b^2}+\frac{(x+a^2)(a-b)(1+c^2)}{x+c^2}=0$$
$$\Leftrightarrow (b-c)(1+a^2)(x+b^2)+(x+a^2)(c-a)(1+b^2)+\frac{(x+a^2)(x+b^2)(a-b)(1+c^2)}{x+c^2}=0$$
$$\Leftrightarrow (b-c)(1+a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/306154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 0
} |
Find $\lim_{n\to \infty}\frac{1}{\ln n}\sum_{j,k=1}^{n}\frac{j+k}{j^3+k^3}.$ Find
$$\lim_{n\to \infty}\frac{1}{\ln n}\sum_{j,k=1}^{n}\frac{j+k}{j^3+k^3}\;.$$
| $$
\frac{1}{\log n}\sum_{j,k=1}^n \frac{j+k}{j^3 +k^3} = \frac{1}{\log n}\sum_{m=1}^n\left(2\sum_{i=1}^m \frac{m+i}{m^3+i^3} - \frac{1}{m^2}\right)
$$
Since
$$
\lim_{n\to\infty} \frac{1}{\log n}\sum_{m=1}^n \frac{1}{m^2} = 0,
$$
we conclude
$$
\lim_{n\to\infty} \frac{1}{\log n}\sum_{j,k=1}^n \frac{j+k}{j^3 +k^3} = 2 \l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/306831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
How many zeros are there at $1000!$ in the base $24$ I know $1000!$ has $\frac{1000}{5}+\frac{1000}{25}+\frac{1000}{125}+\frac{1000}{625}=249$ terminal zeros in decimal notation, but what if we write $1000!$ in base $24$, how many terminal zeros would it have? Is there a way to compute it?
Regards
| The power of $2$ in $1000!$ is
$$a=\left\lfloor \frac{1000}{2}\right\rfloor+\left\lfloor\frac{1000}{4}\right\rfloor+\left\lfloor\frac{1000}{8}\right\rfloor+\left\lfloor\frac{1000}{16}\right\rfloor+\left\lfloor\frac{1000}{32}\right\rfloor+\left\lfloor\frac{1000}{64}\right\rfloor+\left\lfloor\frac{1000}{128}\right\rfloor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/307644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Help with Euler Substitution Let $a < 0$. Find the following indefinite integral by using the third Euler substitution.
$$\int \frac{dx}{(x^2 + a^2) \sqrt{x^2 - a^2}}$$
Where the third Euler substitution is defined by:
Given an integral of the form $\int R(x, \sqrt{ax^2 + bx + c})dx,$ $a \neq 0$.
If the quadratic polyn... | I went with $\sqrt{x^2-a^2}=t (x+a)$, although I doubt it matters. In any case, there's just a bunch of algebra to work through:
$$t=\sqrt{\frac{x-a}{x+a}} \implies x = a \frac{1+t^2}{1-t^2}$$
$$dx = \frac{4 a t}{(1-t^2)^2} dt$$
$$x^2+a^2 = 2 a^2 \frac{1+t^4}{(1-t^2)^2}$$
$$x^2-a^2 = a^2 \frac{4 t^2}{(1-t^2)^2}$$
so t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/308807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Implicit differentiation 2 questions? Hello everyone I have two questions on implicit differentiation.
My first one is express $\frac{dy}{dx}$ in terms of $x$ and $y$ if $x^2-4xy^3+8x^2y=20$
what I did is this
$2x-4x(3y^2)(\frac{dy}{dx})+4y^3+8x^2\frac{dy}{dx}+16xy=0$
$-4x(3y^2\frac{dy}{dx})+8x^2\frac{dy}{dx}=-2x-4y^... | There are slight computation errors and omission errors; Other than that you are doing fine. If you experience such errors, it is better to write every step.
In the first one,
when you differentiate both sides you get $2x-4(x3y^2 dy/dx +y^3)+8(2xy+x^2 dy/dx)=0$. Thus $(-4x3y^2+8x^2)dy/dx=-2x+4y^3-16xy$ so that $dy/dx=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/310408",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How does this sum go to $0$? http://www.math.chalmers.se/Math/Grundutb/CTH/tma401/0304/handinsolutions.pdf
In problem (2), at the very end it says
$$\left(\sum_{k = n+1}^{\infty} \frac{1}{k^2}\right)^{1/2} \to 0$$
I don't see how that is accomplished. I understand the sequence might, but how does the sum $$\left ( \fr... | A related problem. Note that,
$$ \sum_{k=n+1}^{\infty} \frac{1}{k^2}= \sum_{k=1}^{\infty} \frac{1}{(k+n)^2} \leq \sum_{k=1}^{\infty} \frac{1}{k^2} <\infty, $$
which implies that the series
$$ \sum_{k=1}^{\infty} \frac{1}{(k+n)^2} $$
converges uniformly. So, we have
$$ \lim_{n\to \infty} \sqrt{ \sum_{k=n+1}^{\infty}\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/311528",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 8,
"answer_id": 6
} |
Maclaurin series for $\frac{x}{e^x-1}$ Maclaurin series for
$$\frac{x}{e^x-1}$$
The answer is
$$1-\frac x2 + \frac {x^2}{12} - \frac {x^4}{720} + \cdots$$
How can i get that answer?
| An other approach would be to take the inverted first:
$$ \frac{e^x-1}{x} = 1 + \frac{x}{2} + \frac{x^2}{6} +\frac{x^3}{24} + \frac{x^4}{120} \cdots $$
and then you go:
$$\frac{x}{e^x-1} = (\frac{e^x-1}{x})^{-1} = [1 + (\frac{x}{2} + \frac{x^2}{6} +\frac{x^3}{24} +\frac{x^4}{120})]^{-1} $$
and then expand this as in th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/311817",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 4,
"answer_id": 2
} |
Proof of inequality involving logarithms How could we show that
$$\left|\log\left( \left({1 + \frac{1}{n}}\right)^{n + \frac{1}{2}}\cdot \frac{1}{e}\right)\right| \leq \left|\log\left( \left({1 - \frac{1}{n}}\right)^{n - \frac{1}{2}}\cdot \frac{1}{e}\right)\right| ,\; \forall n \text{ sufficiently large?} $$
I already... | The left hand side is
$$
\begin{align}
&\left|\,\left(n+\frac12\right)\log\left(1+\frac1n\right)-1\,\right|\\
&=\left(n+\frac12\right)\left(\frac1n-\frac1{2n^2}+\frac1{3n^3}-\frac1{4n^4}+\dots\right)-1\\
&=\frac1{3\cdot4n^2}-\frac2{4\cdot6n^3}+\frac3{5\cdot8n^4}-\frac4{6\cdot10n^5}+\dots+\frac{(-1)^k(k-1)}{2k(k+1)n^k}+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/312123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
$\int_0^1 x(1-x)\log(x(1-x))dx=?$ I would like to compute the following integral. The answer must be $-\frac{5}{18}$ but I do not know how to evaluate.
$$\int_0^1 x(1-x)\log(x(1-x))dx$$
Thank you for your help in advance.
| Hint:
$$\int_0^1 x^k \log{x} = -\frac{1}{(k+1)^2}$$
Taylor expand the $\log(1-x)$ piece about $x=0$.
$$\begin{align}\int_0^1 x(1-x)\log(x(1-x))dx &= \int_0^1 x(1-x)\log{x} + \int_0^1 x(1-x)\log{(1-x)}\\&= -\frac{1}{4} + \frac{1}{9} - \sum_{k=1}^{\infty} \frac{1}{k}\int_0^1 dx \: x(1-x) x^k \\ &= -\frac{5}{36} - \sum_{k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/312966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Prove that $ \left(1+\frac a b \right) \left(1+\frac b c \right)\left(1+\frac c a \right) \geq 2\left(1+ \frac{a+b+c}{\sqrt[3]{abc}}\right)$.
Given $a,b,c>0$, prove that $\displaystyle \left(1+\frac a b \right) \left(1+\frac b c \right)\left(1+\frac c a \right) \geq 2\left(1+ \frac{a+b+c}{\sqrt[3]{abc}}\right)$.
I e... | Hint: $$\frac{a}{b} + \frac{a}{c} + 1 \ge 3 \frac{a}{\sqrt[3]{abc}}$$ by AM-GM.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/313605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 0
} |
What is the limit of this series? assume $$x_n=\frac{n+1}{2^{n+1}}\sum_{k=1}^n\frac{2^k}{k} ,n=1,2,.....$$
how compute $\lim_{n\to +\infty}x_n$?
Thanks for any hint
| Since the answer is straightforward if we exploit the Cesaro-Stolz theorem, here I want to present a solution that does not use this theorem.
Notice first that whenever $1<r<2$, we have
$$ \lim_{n\to\infty} \frac{\sum_{k＝1}^{n} \frac{r^k}{k}}{\frac{2^{n+1}}{n+1}} ＝ 0.$$
Indeed, the numerator is dominated by $n r^n$, th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/315122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find the Taylor series of $\frac{1}{x+1} $ at $x=2$ This is what I did:
$\begin{align*}
f(x)&=&(x+1)^{-1}\\
f'(x)&=&-(x+1)^{-2}\\
f''(x)&=&2(x+1)^{-3}\\
f'''(x)&=&-6(x+1)^{-4}\\
f''''(x)&=&24(x+1)^{-5}\\
&\vdots\\
f^{(n)}(x)&=&(-1)^nn!(x+1)^{-(n+1)}
\end{align*}$
Then I substituted $x=2$:
$f^{(n)}(2)=(-1)^nn!(2+1)^{-(n... | This is the approach mentioned by Andre:
$$\frac{1}{x+1}=\frac{1}{3+(x-2)}=\frac{1}{3}\frac{1}{1+\frac{x-2}{3}}=\frac{1}{3}\left(1-\frac{x-2}{3}+\left(\frac{x-2}{3}\right)^2-\ldots\right)=$$
$$=\frac{1}{3}\sum_{n=0}^\infty(-1)^n\left(\frac{x-2}{3}\right)^n$$
The above being true whenever
$$\left|\frac{x-2}{3}\right|<1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/316631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Solve $x^{11}+x^8+5\equiv 0\pmod{49}$ Solve $x^{11}+x^8+5\pmod{49}$
My work
$f(x)=x^{11}+x^8+5$
consider the polynomial congruence $f(x) \equiv 0 \pmod {49}$
Prime factorization of $49 = 7^2$
we have $f(x) \equiv 0 \mod 7^2$
Test the value $x\equiv0,1,2,3,4,5,6$ for $x^{11}+x^8+5 \equiv 0\pmod 7$
It works for $x\equiv... | If $x = 1 + 7 t$ for an integer $t$, then $x^{11} \equiv 1 + 11 (7 t) \mod 49$,
$x^8 \equiv 1 + 8 (7 t) \mod 49$, and so $f(x) \equiv 7 + 19 (7t) \mod 49$.
Thus you want $1 + 19 t \equiv 0 \mod 7$, ...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/317458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Solve $a^3-5a+7=3^b$ over the positive integer Solve $a^3-5a+7=3^b$ over the positive integer
I don't know how to solve such equation, please help me. Thanks
| When $b=1$, we already know the result.
I can prove when $3\mid b$, there is no solution.
You may see
if we try $a = 3^{b/3}$, then
$a^3-5a+7 = 3^b-5\cdot3^{b/3}+7<3^b$, if $b\ge3$.
If we try $a = 3^{b/3}+1$, then
$a^3-5a+7 = 3^b+3\cdot 3^{2b/3}-2\cdot 3^{b/3}+3>3^b$, if $b\ge 3.$
Thus $3^{b/3}<a<3^{b/3}+1$, there is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/317860",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 2
} |
Find point closest to the given point I need to find the closest point of the graph $y=\sqrt x$ to the point $(4,0)$.
This is how I understand it:
I assume that triangle with the sides $a$, $x$ and $x/2$ and the largest triangle are similar triangles. From that I conclude that the smaller triangle's one side is $x/2$.... | The distance from a point on the graph $y=\sqrt{x}$ to the point $(4,0)$ is $$\sqrt{(x-4)^2+(\sqrt{x}-0)^2}.$$ This will be minimized precisely when $$(x-4)^2+(\sqrt{x}-0)^2=x^2-8x+16+x=x^2-7x+16$$ is minimized, since $\sqrt{\cdot}$ is an increasing function. Do you know how to find $x\geq 0$ that minimizes $x^2-7x+16$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/318117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Understanding a particular evaluation of $\prod\limits_{n=2}^{\infty}\left(1-\frac{1}{n^3}\right)$ I'm having a hard time understanding the following evaluation of the infinite product $$ \prod_{n=2}^{\infty}\left(1-\frac{1}{n^3}\right).$$
In particular, I don't understand how you go from line 2 to line 3.
Here $\omeg... | Here's my understanding of that evaluation.
$$ \begin{align} \prod_{n=2}^{\infty}\left(1-\frac{1}{n^3}\right) &= \prod_{n=2}^{\infty}\frac{(n-1)(n^2+n+1)}{n^3} \\ & = \prod_{n=2}^{\infty} \frac{(n-1)(n- \omega)(n-\omega^{2})}{n^{3}} \\ &= \lim_{m \to \infty} \prod_{n=2}^{m} \frac{n-1}{n} \frac{(n- \omega)(n-\omega^{2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/319784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
"answer_count": 1,
"answer_id": 0
} |
For this matrix $A$, what is $A^n$? $$A = \begin{pmatrix}0 & a & b \\ 0& 0 & c \\ 0& 0 &0\end{pmatrix}$$
What is $A^n$ (for $n\geq 1)$?
| This matrix is a nilpotent one, so we know that $A^3=0$. Now we calculate
$$A^2=A\cdot A = \begin{pmatrix} 0 & a & b \\ 0 & 0& c \\0 & 0 & 0 \end{pmatrix}\cdot
\begin{pmatrix} 0 & a & b \\ 0 & 0& c \\0 & 0 & 0 \end{pmatrix}=
\begin{pmatrix} 0 & 0 & ac \\ 0 & 0& 0 \\0 & 0 & 0 \end{pmatrix}$$
As $A^3=0$ $A^n=0$ for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/321360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Algorithm for finding the square root of a polynomial... I'm going through Wallace Clarke Boyden's A First Book in Algebra, and there's a section on finding the square root of a perfect square polynomial, eg. $4x^2-12xy+9y^2=(2x-3y)^2$. He describes an algorithm for finding the square root of such a polynomial when it'... | It's just saying start with the highest degree and work down. This may not be the fastest but will work or tell you that the thing is not really a square. So, begin with $x^3,$ since the square must be $x^6$ and we get one free choice, $\pm x^3.$ Next, $(x^3 + A x^2)^2 = x^6 + 2 A x^5 + \mbox{more}.$ So $2A = -2, A = -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/324385",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 2,
"answer_id": 1
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Help determining the parameterized solution to a system of linear equation. I am quite new to this area so please bear with me if I am overlooking something glaringly obvious here :)
I am trying to solve the following equation system:
$$\begin{array}{lcl} x + y + z & = & 30 \\ x + 2y & = & 25 \\ 2x + 3y +x& = & 55 \end... |
From this development is is much more clear that z should be the parameter since x,y can be expressed as linear functions of z?
Precisely. $z$ is a "free" variable (as you can tell by the last row of zeros in both approaches) on which the precise values of $x, y$ depend. Once a value for $z$ is chosen (among infinite... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/324960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Intersection points of a Triangle and a Circle How can I find all intersection points of the following circle and triangle?
Triangle
$$A:=\begin{pmatrix}22\\-1.5\\1 \end{pmatrix} B:=\begin{pmatrix}27\\-2.25\\4 \end{pmatrix} C:=\begin{pmatrix}25.2\\-2\\4.7 \end{pmatrix}$$
Circle
$$\frac{9}{16}=(x-25)^2 + (y+2)^2 + (z-3)... | If $\,\;\;\vec a+t\vec b\,,\;t\in\Bbb R\;\;$ is a line in $\,\Bbb R^3\,$, then each of its points can be expressed as
$$\begin{pmatrix}\alpha_1:=a_1+tb_1\\\alpha_2:=a_2+tb_2\\\alpha_3:=a_3+tb_3\end{pmatrix}\;,\;\;a_i,b_i,t\in\Bbb R$$
then a point as above belongs to the circle
$$(x-25)^2+(y+2)^2+(z-3)^2=9\iff (\alpha_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/325411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Need help proving this integration If $a>b>0$, prove that :
$$\int_0^{2\pi} \frac{\sin^2\theta}{a+b\cos\theta}\ d\theta = \frac{2\pi}{b^2} \left(a-\sqrt{a^2-b^2} \right) $$
| You can solve it with direct integration some subsitutions gives you the antiderivative:
$$\frac{-2 \sqrt{b^2-a^2} \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{\theta }{2}\right)}{\sqrt{b^2-a^2}}\right)+a \theta -b \sin (\theta )}{b^2}$$
use a substituion like
$t=\tan(\theta)$. (and a lot of the trigonometric ident... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/326714",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
Solve $x^{10} + 4x^3 +3x + 4 \equiv 0 \pmod {(4\cdot3)}$ Solve $x^{10} + 4x^3 +3x + 4 \equiv 0 \pmod {(4\cdot3)}$
Work:
Let $P(x) = x^{10} +4x^3+3x+4 \equiv 0 \pmod 2$ and $12=4\cdot3=2^2\cdot3$
We have $P(x) \equiv0 \pmod {2^2}$ and $P(x)\equiv0\pmod3$
For $\pmod {2^2}$,
$x^{10} + 4x^3 +3x + 4 \equiv 0 \pmod 2$
Try $... | If you want $n \equiv 1 \pmod 3$ and $n \equiv 0 \pmod 4$ you can just try the multiples of $4$ until you find one that works. In this case it is $4$. So $P(4)\equiv 0 \pmod {12}$. Here is a check from Alpha
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/328090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Representing Functions as Power Series Rewrite $$f(x)=(1+x)/(1-x)^2$$ as a power series.
Work thus far:
I separated it into two parts:
$$1/(1-x)^2 + x/(1-x)^2$$
I realize that the first expression is the derivative of $1/(1-x)$ and come up with this sum of series:
$$\sum_{n=0}^\infty x^n$$
Since a derivative was involv... | $$
\frac{1}{1-x}=1+x+x^2+x^3+\ldots
$$
$$
\frac{1}{(1-x)^2}=1+2x+3x^2+4x^3+\ldots
$$
$$
\frac{1+x}{(1-x)^2}=1+2x+3x^2+\ldots +x+2x^2+ 3x^3+\ldots=1+3x+5x^2+7x^3+\ldots
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/329070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
How to compute $\int_a^b (b-x)^{\frac{n-1}{2}}(x-a)^{-1/2}dx$? I wonder how to compute the following integral (for any natural number $n\geq 0$):
$$\int_a^b (b-x)^{\frac{n-1}{2}}(x-a)^{-1/2}dx.$$
Does anyone know the final answer and how to get there? Is there a trick?
Thank you very much!
| $$I=\int_a^b\frac{\sqrt{b-x}^{\,n}}{\sqrt{(b-x)(x-a)}}\,dx$$
$$x=a\cos^2 t+b\sin^2 t:$$
$$\begin{align*}I&=\int_0^{\pi/2}\frac{2(b-a)\cos t\sin t\cdot\sqrt{(b-a)\cos^{2}t}^{\,n}}{\sqrt{(b-a)^2\cos^2 t\sin^2 t}}\,dt\\[7pt]&=2\sqrt{b-a}^{\,n}\int_0^{\pi/2}\cos^{n}t\,dt\\[7pt]&= \left\{
\begin{array}{l l}
\pi\sqrt{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/329561",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
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Derive the polynomial Given that the solutions to a cubic equation using Cardano's method are $$x_1, x_2, x_3=\sqrt[3]{-\frac{260}{9}i\sqrt{3}-21}+\sqrt[3]{\frac{260}{9}i\sqrt{3}-21}-3$$ derive the cubic polynomial and its factors using algebraic methods only, i.e. without using trigonometric functions.
My initial thou... | A way to get to the cube roots of the nested radicals in the question, thus getting to the three separate roots "buried" in the Cardano formula above, without using DeMoivre's formula to find the cube roots of the nested radicals is, first set
$x=-21$ and $y=-\frac{260}{9}i\sqrt{3}$ and (without going into how the equ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/331601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How to solve this system of equations? How to solve this system of equations?
$$\begin{cases}
1+\sqrt{2 x+y+1}=4 (2
x+y)^2+\sqrt{6 x+3 y},\\
(x+1) \sqrt{2 x^2-x+4}+8 x^2+4
x y=4.
\end{cases}$$
| Hint
Define $U=2x+y$ in first equation
You will get
$1+\sqrt{U+1}=4 U^2+\sqrt{3U}$ solve $U$
here you need to solve $U$
$$(x+1) \sqrt{2 x^2-x+4}+8 x^2+4xy=4$$
$$(x+1) \sqrt{2 x^2-x+4}+4x(2x+y)=4$$
Then put U in second equation and find $x$
$$(x+1) \sqrt{2 x^2-x+4}+4xU=4$$
after finding $x$ , you can get $y$ from $U... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/332059",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How to find $\lim_{x\to0^+}\frac{1 - \frac{2}{\pi} \arcsin(_{2}F_{1}(1/2, 1/2, 3/2, x^4))}{x^2}$? Does the limit exist?
$$\lim_{x\to0^+}\frac{1 - \displaystyle \frac{2}{\pi} \arcsin(_{2}F_{1}(1/2, 1/2, 3/2, x^4))}{x^2}$$
| We may use a classical hypergeometric identity (equation $(14)$ from Cook's 'Notes on hypergeometric functions') :
$$_{2}F_{1}\left(\frac 12, \frac 12; \frac 32; z^2\right)=\frac {\arcsin(z)}{z}$$
to rewrite your limit as :
$$\lim_{x\to0^+}\frac{1 - \displaystyle \frac{2}{\pi} \arcsin\left(\frac {\arcsin\left(x^2\right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/332141",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Simplify $ \frac{1}{x-y}+\frac{1}{x+y}+\frac{2x}{x^2+y^2}+\frac{4x^3}{x^4+y^4}+\frac{8x^7}{x^8+y^8}+\frac{16x^{15}}{x^{16}+y^{16}} $ Please help me find the sum
$$
\frac{1}{x-y}+\frac{1}{x+y}+\frac{2x}{x^2+y^2}+\frac{4x^3}{x^4+y^4}+\frac{8x^7}{x^8+y^8}+\frac{16x^{15}}{x^{16}+y^{16}}
$$
| $$
\left(\left(\left(\left(\left(\frac{1}{x-y}+\frac{1}{x+y}\right)+\frac{2x}{x^2+y^2}\right)+\frac{4x^3}{x^4+y^4}\right)+\frac{8x^7}{x^8+y^8}\right)+\frac{16x^{15}}{x^{16}+y^{16}}\right)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/332191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 4
} |
Calculating $\lim_{x \rightarrow 1}(\frac{23}{1-x^{23}} - \frac{31}{1-x^{31}})$ How to calculate following limit?
$$\lim_{x \rightarrow 1}\left(\frac{23}{1-x^{23}} - \frac{31}{1-x^{31}}\right)$$
| Generalized form
$$\lim_{x \rightarrow 1}\left(\frac{m}{1-x^{m}} - \frac{n}{1-x^{n}}\right) = \frac{n-m}{2}$$
$Proof:$
$Let$ $L = \lim_{x \rightarrow 1}\left(\frac{m}{1-x^{m}} - \frac{n}{1-x^{n}}\right)$
$Let x = \frac{1}{y}$
$if x\rightarrow 1$ $then$ $y\rightarrow 1$
$L =\lim_{x \rightarrow 1}\left(\frac{m}{1-... | {
"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": 2
} |
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