Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Show that $2\cos(x)$ is equal to $2\cos(2x)\sec(x)+\sec(x)\tan(x)\sin(2x)$ This is from the derivative of $\dfrac{\sin(2x)}{\cos x}$
I tried to solve it and arrived with factoring the $\sec(x)$ but I still can't get it to $2\cos(x)$. Could you help me out, please? Thanks
| Using Double angle formulae, $$\sec x\left(2\cos2x+\tan x\sin2x\right)$$
$$=\sec x\left[2(2\cos^2x-1)+\frac{\sin x}{\cos x}\cdot2\sin x\cos x\right]$$
$$=\sec x\left[4\cos^2x-2+2\sin^2x\right]$$
$$=\sec x\left[4\cos^2x-2(1-\sin^2x)\right]$$
$$=\sec x\left[2\cos^2x\right]=?$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/972819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Find order of group given by generators and relations Let $G$ be the group defined by these relations on the generators $a$ and $b$: $\langle a, b; a^5, b^4, ab=ba^{-1}\rangle$. I need hints how to find order of $G$.
| The only good way to do problems like this is to play around with words in the letters $a,b,a^{-1},b^{-1}$, seeing if you can use those given relations to get any word in some standard form. Then, you can say every element in $G$ is equivalent to one in this form, and there are so many words in that form, so $G$ has th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/972873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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Modify the closest-pair algorithm to use the $L_\infty$ distance. I'm trying to understand the closest pair of points problem. I am beginning to understand the two-dimensional case from a question a user posted some years ago. I'll link it in case someone wants to look at it: For 2-D case (plane) - "Closest pair of poi... | The only difference is in the "merging" part of the recursion. Let the two sets created by the dividing line be called as $L$ and $R$, respectively (for left and right sets). Via recursion, we have our temporary current closest pair -- let's say its at distance $M$. Similar to the original algorithm, we filter only the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/972993",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Calculating future data based previous data
The sales volume of the next month is predicted by the data in the
past. The sales volume is changed greatly from month to month, but
the annual fluctuation pattern is almost the same every year. Which
of the following is the most appropriate formula that can be... | Answer d may be the best option, since it calculates the mean of the values observed in the successive month during the previous 3 years. The information that sales volume changes greatly from month to month, but with the same annual fluctuation pattern every year, suggests that a reliable prediction can be based on th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/973061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Verify the divergence theorem for a sphere
Question i cannot work out. I assume you need to get both sides in terms of u and v (parameterized), but im getting pretty confused after completing the first few steps.
| The left-hand size is over the solid ball $V$, whereas the right-hand side is over just its boundary, the sphere of radius $3$ centered at the origin. The fact that we can translate an integral in two dimensions into one in three dimensions (which may be easier) is what makes the Divergence Theorem a powerful tool.
For... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/973149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Help with an inequality involving a convex function Let $a< f(x) < b $, $x \in \Omega $, $\mu(\Omega )=1 $, and set $t=\int f d \mu $.
Then $a < t < b $.
Suppose $\phi $ is a convex function on $(a,b) $
then by definition of convexity we have that for $a<s<t<u<b $, $\frac {\phi (t)- \phi(s)} {t-s } <\frac {\phi (u)-... | Since
$$ \beta = \sup_{s\in(a,t)} \frac{\phi(t)-\phi(s)}{t-s} $$
we can take $s=u$ in the case $u\in(a,t)$ to get
$$ \beta \ge \frac{\phi(t)-\phi(u)}{t-u} $$
Since $t-u>0$ in this case, this yields
$$ (t-u)\beta \ge \phi(t)-\phi(u) $$
Rearranging yields the desired inequality.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/973232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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problem using Weierstrass-Approximation
Prove that $$\int_0^1 f(x)x^ndx=\frac{1}{n+2}$$ for each $n=0,1,2,\cdots \implies$ $f(x)=x$ on $[0,1]$
my attempt: for some sequence of coefficients $(a_n)$, choose some polynomial $p_n(x)=a_0+a_1x+...+a_nx^n$ such that $p_n\to f$ uniformly by Weierstrass-Approximation. Then $... | Your assumptions imply that
$$\int_0^1 (f(x)-x)x^ndx=0$$
for all $n$. Now deduce that $f(x)-x=0$. The standard way to do that is to approximate $f(x)-x$ with a polynomial sequence $p_n(x)$ and then deduce that
$$\int_0^1 (f(x)-x)^2dx=0$$
since
$$\int_0^1 (f(x)-x)p_n(x)dx=0$$
for each $n$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/973336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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what is the derivative of $3\cos(\cos x)\;?$ what is the derivative of $3\cos(\cos x)\;?$
I think I need to use the chain rule and i believed it to be $3-\sin(\cos x)(-\sin x)$ but this is not the case.
| You applied the chain rule correctly if you meant to write the derivative as a product of three factors, i.e. $$\dfrac{dy}{dx} = 3(-\sin(\cos x))\cdot(-\sin x).$$ You just need to simplify to get $$--3\sin(\cos x)\cdot \sin x = 3\sin(\cos x)\cdot \sin x$$ which can also be written $$3(\sin x)\sin(\cos x)$$
Writing it a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/973413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Solvable subgroups in $GL(n,F)$ Is it true, that any solvable subgroup $G$ in $GL(n,F)$ is subgroup of upper triangular matrix in some basis?
| If $F$ is algebraically closed and the group is connected ( and algebraic) then the answer is yes. This is Borel's theorem -- the search term is Borel subgroup.
However, we need connected, even for $F$ algebraically closed (@Derek Holt: thanks for pointing the example of finite solvable groups ) Indeed, any (solvabl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/973520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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To show a finite group G is nilpotent Let G be a finite group and G' denotes it's commutator.If order of G' is 2.Then show that G is Nilpotent.
What I have tried:G/G' is abelian,so it is nilpotent again G' is nilpotent as it's order is 2.But from this I can't conclude that G is nilpotent.So I am trying to show that all... | You need to show that the derived subgroup $G^\prime$ is central. To this end, take any commutator $[a,b]$ and any $g\in G$. Since $G^\prime$ is normal, and has order $2$, it must be that $g^{-1}[a,b]g = [a,b]$, so $[a,b]$ commutes with $g$. And, since $g$ was arbitrary, it follows that $G^\prime\leq Z(G)$. Thus, $G... | {
"language": "en",
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A problem on almost sure convergence of an average I have the following exercise:
Let $X_1, X_2 \ldots$ be such that $$ X_n = \left\{
\begin{array}{ll} n^2-1 & \mbox{with probability } n^{-2} \\ -1
& \mbox{with probability } 1-n^{-2} \end{array} \right. $$ Show that
$S_n/n \rightarrow -1$ almost surely.
... | Hint: Apply the Borel-Cantelli lemma to the sequence of events
$$A_n := \{\omega \in \Omega; X_n(\omega) \neq -1\}, \qquad n \in \mathbb{N}.$$
| {
"language": "en",
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Given $f(x)= e^x - e^ax$ with roots $P$ and $Q$,$0I have a midterm tomorrow and while I was looking through old exams from my professor I stumbled on a problem for which I'm not able to see the solution.
We want to find the rots of $f(x) = e^x - xe^a$ with $a>1$.
Consider the fixed point functions $g_1(x) = e^x/e^a$ an... | Notice that
$$f(P)=0\Rightarrow e^P-e^aP=0\Rightarrow e^P=e^aP\Rightarrow P=\frac{e^P}{e^a}=g_1(P)$$
In the same way for $Q$. So $g_1(x)$ has two fixed points.
Now
$$f(P)=0\Rightarrow e^P-e^aP=0\Rightarrow e^P=e^aP\Rightarrow P=a+\ln(P)=g_2(P)$$
The same technique for $Q$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The rationals as a direct summand of the reals The rationals $\mathbb{Q}$ are an abelian group under addition and thus can be viewed as a $\mathbb{Z}$-module. In particular they are an injective $\mathbb{Z}$-module. The wiki page on injective modules says "If $Q$ is a submodule of some other left $R$-module $M$, then t... | Consider the short exact sequence of $\mathbb Z$-modules
$$
0 \to \mathbb Q \hookrightarrow \mathbb R \xrightarrow{\varphi} \mathbb R / \mathbb Q \to 0.
$$
Since $\mathbb Q$ is injective, the sequence splits. Thus, there exists a homomorphism $\lambda: \mathbb R / \mathbb Q \to \mathbb R$ such that $\varphi \circ \lamb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/974011",
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Am I assuming too much in this Natural Deduction proof? So I need to prove the following using natural deduction:
$M \to J, A \to J, \lnot M \to A, A \to \lnot J \vdash M, J, \lnot A$
This is my proof so far:
1.) $M \to J$
2.) $A \to J$
3.) $\lnot M \to A$
4.) $A \to \lnot J$
5.) $(M \to J) \lor (A \to J) ----(\lor I ... | From :
2) $A→J$
and
4) $A→¬J$
assuming : [a] $A$
we get $J$ and $\lnot J$ and thus, by $\land$-I, a contradiction :
$J \land \lnot J$.
Thus, by $\lnot$-E followed by $\lnot$-I we derive :
$\lnot A$
discharging [a].
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Comparing cardinalities Why these two sets are equinumerous?
$$[0,1]^\Bbb N\text{ and }\Bbb Q^\Bbb N$$
Here is my reason:
The set of rational numbers $\Bbb Q$ is countably infinite. However, $[0, 1]$ is not countable and is infinite.
So, they shouldn't be equinumerous.
Even, there is the power of $\Bbb N$, it shouldn'... | First of all, note that $\Bbb{Q^N}$ includes $\{0,1\}^\Bbb N$, so it too is uncountable. But just being uncountable doesn't mean much because there are uncountable sets of different cardinalities.
But note that $|[0,1]|=2^{\aleph_0}$ and $|\Bbb Q|=\aleph_0$. Therefore $[0,1]^\Bbb N$ has cardinality $(2^{\aleph_0})^{\al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/974293",
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What is inverse of $I+A$ given that $A^2=2\mathbb{I}$? I have the next problem:
Let $A$ be a real square matrix such that $A ^ 2 = 2\mathbb{I}$. Prove that $A +\mathbb{I}$ is an invertible matrix and find its inverse.
I tried with the answers given here:What is inverse of $I+A$?
Any hints?
| Note that
$(\Bbb I + A)(\Bbb I - A) = \Bbb I - A^2 = \Bbb I - 2 \Bbb I = -\Bbb I, \tag{1}$
or
$(\Bbb I + A)(A - \Bbb I) = \Bbb I. \tag{2}$
(2) shows both that $(\Bbb I + A)^{-1}$ exists and that it is given by
$(\Bbb I + A)^{-1} = A - \Bbb I; \tag{3}$
without further knowledge of $A$, not much more can be said. One ca... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Does $x \perp (y,z)$ imply $x \perp y \mid z$? Does $x \perp (y,z)$ imply $x \perp y \mid z$, where $\perp$ denotes stochastic independence?
I was told it is true and the following is the proof (which I believe is wrong):
We want to show that $p(x,y,z) = p(x)p(y,z)$ implies $p(x,y \mid z)=p(x \mid z) p(y \mid z)$.
then... | The proof is correct and it shows the desired result.
To see this, sum on $y$ the identity $p(x,y\mid z)=p(x)p(y\mid z)$, valid for every $(x,y,z)$. One gets $p(x\mid z)=\sum\limits_yp(x,y\mid z)=p(x)\sum\limits_yp(y\mid z)=p(x)$ for every $(x,z)$ hence $p(x)=p(x\mid z)$ for every $(x,z)$.
Thus, the identity $p(x,y\m... | {
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"timestamp": "2023-03-29T00:00:00",
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Supremum of a set of irrational numbers I need help with the following example:
Let S be the set of all irrationals in $[0,1]$. Show that $\sup(S) = 1$.
Is there some property that I should be referring to when proving problems like these? The set definition states that it is bounded between $[0,1]$ so is it possible t... | Note that $1$ is an upper bound for the set $S.$ We must show that is it the least upper bound.
(It means we have to show that any thing less than $1$ cannot be an upper bound.)
Take any $\epsilon>0.$
Then we can find a large natural number $n>1$ such that $\dfrac{1}{n}<2\epsilon.$
This gives us $$1-2\epsilon<1-\dfrac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/974559",
"timestamp": "2023-03-29T00:00:00",
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Is there a faster way to add/subtract fractions then having to draw a factor tree each time? Do you really have to draw a factor tree and work with primes every time you encounter adding or subtracting fractions?
Not this way - LCM(8,15)...
15: 15, 30, 45, 50, 65, *80* --
8: 8, 16, 24, 32, 40, 48, 56, 64, 72, *80* -- ... | We don't need the least common multiple to add fractions.
But if you want the least common multiple (lcm) of $x$ and $y$, where $x$ and $y$ are BIG, first use the Euclidean Algorithm to find the greatest common divisor $\gcd(x,y)$ efficiently. Then use the fact that $\operatorname{lcm}(x,y)=\frac{xy}{\gcd(x,y)}$.
For v... | {
"language": "en",
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Can a limit of form$\ \frac{0}{0}$ be rational if the numerator is the difference of transcendental functions, and the denominator a polynomial one? Let$\ f_1(x)$ and$\ f_2(x)$ be transcendental functions such that$\ \lim_{x\to 0} f_1(x)-f_2(x)=0$, and$\ f_3(x) $ polynomial, such that$\ f_3(0)=0$. Can$\ \lim_{x\to 0} \... | $$
\frac{\sin x -(e^x-1)}{x^2} \to - \frac12
$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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The smallest $n$ for which the sum of binomial coefficients exceeds $31$ I have a problem with the binomial theorem.
What is the result of solving this inequality:
$$
\binom{n}{1} + \binom{n}{2} + \binom{n}{3} + \cdots +\binom{n}{n} > 31
$$
| Since we have
$$\sum_{k=0}^{n}\binom{n}{k}=\sum_{k=0}^{n}\binom{n}{k}\cdot 1^{n-k}\cdot 1^k=(1+1)^n=2^n,$$
we have
$$\sum_{k=1}^{n}\binom{n}{k}\gt 31\iff\sum_{k=\color{red}{0}}^{n}\binom{n}{k}\gt 32\iff 2^n\gt 2^5\iff n\gt 5.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/974848",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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If $x=t^2\sin3t$ and $y=t^2\cos3t$, find $\frac{dy}{dx}$ in terms of $t$ If $x=t^2\sin3t$ and $y=t^2\cos3t$, find $\frac{dy}{dx}$ in terms of $t$. This is how I tried solving it:
$$
\frac{dx}{dt} = 2t\sin3t + 3t^2\cos3t \\
\frac{dy}{dt} = 2t\cos3t - 3t^2\sin3t \\
\frac{dy}{dx} = \frac{2t\cos3t - 3t^2\sin3t}{2t\sin3t + ... | You are on the right trace. Just divide $t\cos 3t$ in the numerator and denominator of $\frac{dy}{dx}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/974960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Curvature of curve not parametrized by arclength If I have a curve that is not parametrized by arclength, is the curvature still $||\gamma''(t)||$? I am not so sure about this, cause then we don't know that $\gamma'' \perp \gamma'$ holds, so the concept of curvature might not be transferable to this situation. So is th... | Here is sort of a simple example. Consider $\lambda_1,\lambda_2>1$ with $\lambda_1\not=\lambda_2$. Consider the unit disk. Both of the curves $(\cos(\lambda_1 t),\sin(\lambda_1t))$ and $(\cos(\lambda_2 t),\sin(\lambda_2t))$ trace it out. It makes intuitive sense to define the curvature as the rate of change of the velo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/975062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
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Solving recurrence -varying coefficient How can one find a closed form for the following recurrence?
$$r_n=a\cdot r_{n-1}+b\cdot (n-1)\cdot r_{n-2}\tag 1$$
(where $a,b,A_0,A_1$ are constants and $r_0=A_0,r_1=A_1$)
If the $(n-1)$ was not present, this could easily be solved using a characteristic equation. However, wit... | The e.g.f.'s $\sum_{n=0}^\infty r_n t^n/n!$ for two linearly independent solutions
are
$ \exp(b t^2/2 + a t)$ and $\exp(b t^2/2 + a t)\; \text{erf}(\sqrt{b/2} t + a/\sqrt{2b})$.
From the first, we get
$$ r_n = n! \sum_{k=0}^{\lfloor n/2 \rfloor} \dfrac{ (b/2)^k a^{n-2k}}{k! (n-2k)!}$$
EDIT: Here's a little explanati... | {
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Doubt on understanding continuity . Just preparing for my multivariable-calculus exam and wanted to clear these things:
I've come across many questions of sort below ,especially 2-dimensional regions, and wanted to understand the Idea behind them....
Prove the continuity of $f(x,y)$ on $\mathbb R^2$where,
$$f(x,y) =... | Step 1: prove that the function is continuous whenever at a point $(x_0,y_0)$ whenever $x_0\ne y_0$. Should be evident since the function is a composition of continuous functions.
Step 2: prove that the function is also continuous at the points $(x_0,x_0)$ for arbitrary $x_0$. Should be also doable. Hint: $$|f(x,y)-f(x... | {
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Does $\lim \frac {a_n} {b_n}$ exist and $\lim a_n \neq 0$ imply $\lim b_n$ exist? Suppose $\lim_{n \rightarrow \infty} \frac {a_n} {b_n}$ exist and $(a_n)$ converges to some number $k \neq 0$. Is it then possible to conclude that $(b_n)$ converges ?
Also, suppose $\lim_{n \rightarrow \infty} \frac {a_n} {b_n}$ exist a... | For the first statement you might try to do:$\lim\frac{a_n}{\frac{a_n}{b_n}}$ but if you consider the case where: $a_n=1+\frac 1n$ and $b_n=n$ then $\lim\frac{a_n}{b_n}=\lim \left( \frac 1n+\frac 1{n^2}\right)=0$ and $\lim a_n=1$ but $\lim b_n=\infty$
And for the second part:$\frac{a_n}{b_n}b_n=a_n$
$\frac{a_n}{b_n}$ i... | {
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Show that $\Gamma \cup \{\neg \phi\}$ is satisfiable if and only if $\Gamma\not \models \phi$
Let $\Gamma$ be a set of formulas and $\phi$ be a formula. Show that $\Gamma \cup \{\neg \phi\}$ is satisfiable if and only if $\Gamma\not \models \phi$.
This seemed pretty obvious but I wanted to see if my proof made sense:... | Your proof is entirely correct. Cheers :).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/975470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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For small $x$, one has $\ln(1+x)=x$? What does it mean that for small $x$, one has $\ln(1+x)=x$? How can you explain this thing ? Thanks in advance for your reply.
| Take the tangent line at of $f(x) = \ln(1+x)$ in $x = 0$.
\begin{align*}
f(x) & \approx f(0) + f'(0) (x - 0) \\
& = \ln(1+0) + \left[\frac{d}{dx} \ln(1+x)\right]_{x = 0} (x-0) \\
& = 0 + 1 x \\
& = x
\end{align*}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/975565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Nice derivation of $\sum_{n=1}^\infty \frac{1}{n} \left( \frac{q^{2n}}{1-q^n}+\frac{\bar q^{2n}}{1-\bar q^n}\right)=-\sum_{m=2}^\infty \ln |1-q^m|^2$ I'm searching for a nice derivation of the formua
$\sum_{n=1}^\infty \frac{1}{n} \left( \frac{q^{2n}}{1-q^n}+\frac{\bar q^{2n}}{1-\bar q^n}\right)=-\sum_{m=2}^\infty \ln ... | Consider the double sum:
$$\sum_{n=1}^\infty \frac{1}{n} \frac{q^{2n}}{1-q^n} = \sum_{n=1}^\infty \frac{1}{n} \sum_{m=0}^\infty q^{n(m+2)}
= \sum_{m=0}^\infty \sum_{n=1}^\infty \frac{1}{n} q^{n(m+2)} = \sum_{m=\color{red}{\mathbf{2}}}^\infty \log (1 - q^{m}) $$
This is a physics paper. They aren't worried that both s... | {
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License plate combination California's license plate is made up for a number, followed by 3 letters, and 3 more numbers. If you cannot have the word BOB then how many license plate can be made in total?
I'm guessing it's $10^4 * 26^3 - 10^4$ because the word BOB is disallowed so any combinations that contain that word... | Your initial guess is correct. Also, note that $10^4(26^3-1) = 10^4\cdot 26^3 - 10^4.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/975898",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Sum of roots of an equation $\sqrt{x-1}+\sqrt{2x-1}=x$ Find the sum of the roots of the equation $\sqrt{x-1}+\sqrt{2x-1}=x$
My attempt: Squaring the equation: $(x-1)+(2x-1) +2\sqrt{(x-1)(2x-1)}=x^2$
$\implies x^2-3x+2=2\sqrt{(x-1)(2x-1)} $
$\implies (x-1)(x-2)=2\sqrt{(x-1)(2x-1)} $
$\implies (x-2)=2\sqrt{\displaystyle ... | The problem is that you divided by $x-1$, so, you lost a root.Strating from $$ (x-1)(x-2)=2\sqrt{(x-1)(2x-1)}$$ as you properly wrote and squaring $$(x-1)^2(x-2)^2=4{(x-1)(2x-1)}$$ Expanding and grouping leads to $$x^4-6 x^3+5 x^2=0$$ so the sum of the roots is $6$ (you can check that the roots are $0,0,1,5$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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What is the correct answer to this diffferential equation? [Question]
When solving the differential equation:
$$\frac{\mathrm dy}{\mathrm dx} = \sqrt{(y+1)}$$
I've found two ways to express $y(x)$:
implicitly: $2\sqrt{(y + 1)} = x + C$
or directly: $y = (x^2)/4 + (2xC)/4 + (C^2)/4 -1$
Although they look the same, the... | The version with $\sqrt{8}$ is the only correct one. If you use $-\sqrt{8}$, you get that $\sqrt{y+1}$ is negative when $x=0$. But $\sqrt{y+1}$, by definition, is the non-negative number whose square is $y+1$.
Remark: A familiar related fact is that when we are solving an ordinary equation that involves square roots, ... | {
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"timestamp": "2023-03-29T00:00:00",
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Evaluating the double limit $\lim_{m \to \infty} \lim_{n \to \infty} \cos^{2m}(n! \pi x)$ I have to find out the following limit $$\lim_{m\to\infty}\lim_{n\to\infty}[\cos(n!πx)^{2m}]$$ for $x$ rational and irrational. for $x$ rational $x$ can be written as $\frac{p}{q}$ and as $n!$ will have $q$ as its factor the limit... | For $m>0$ the limit of $cos^{2m}(n!\pi.x)$ as $n$ goes to infinity does not exist for some $x$. For example, for natural number k, let $f(k)=1/2$ if $k$ is an integer power of $2$, otherwise let $f(k)=0$. Let $x=f(0)/0!+f(1)/1!+f(2)/2!+...$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Consider the family of lines $a(3x+4y+6)+b(x+y+2)=0$ Find the equation....... Question :
Consider the family of lines $a(3x+4y+6)+b(x+y+2)=0$ Find the equation of the line of family situated at the greatest distance from the point P (2,3)
Solution :
The given equation can be written as $(3x+4y+6)+\lambda (x+y+2)=0$ ... | Find the common point of intersection of this family of lines. Here it is (-2,0). Now, given point is (2,3). Equation of line passing through both the points is : 3x + 6 =4y. You actually only need the slope of this line which is : 3/4 Line perpendicular to this line passing through the common point will be at the grea... | {
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"source": "stackexchange",
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Find a function given its poles, residues, limit at infinity, and additional constraints So what is given is that the function f(z) holomorphic is on the Complex plane provided:
*
*$f(z)$ has a first order pole in $z = 1$
*$f(z)$ has a second order pole in $z = 0$ with residue $0$
*$\lim\limits_{z\to\infty} f(z)= ... | Q) the only singularity of a single valued function f(z) are poles of order 2 and 1 at Z=1 and Z=2 with residues of these poles 1 and 3 respectively.if f(0)=3/2 and f(-1)=1,
determine the function f(z)
| {
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About strict inequality in Groups Locally Nilpotent Let $G$ be a locally nilpotent group and $x \in G$. How can I prove that $[G,x] \neq G$?
Note that if $G$ is a nilpotent group then this statement is true, because $G'=[G,G]<G$ and $[G,x]\leq G'$.
However, $G$ can be a locally nilpotent group such that $G=G'$ that thi... | If $G$ is nilpotent and has central series $1 <G_1 < \cdots < G_n=G$ and $x \in G_{i+1} \setminus G_i$ for some $i$, then $[G,x] \le G_i$, and so $x \not\in [G,x]$.
Now suppose that $G$ is locally nilpotent and $x \in [G,x]$. Then $x$ is a finite product of elements of the form $[g_i,x]^{\pm 1}$. But the finitely many ... | {
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Why do we care about the 'rapidness' for convergence? It is those puzzeling improper integrals that I can't get my head around....
Does the (improper) integral $\frac 1{x^2}$ from 1 to $\infty$ coverges because it is converging "fast" or because it has converging anti-derivative? in case of the former, what do you mean... | The comparison tests (either direct or via a limit) are concrete, rigorous statements expressing the intuition behind the statement that
$$\int_a^{\infty} f(x) \, dx$$
converges if $f(x)\rightarrow 0$ as $x\rightarrow\infty$ "fast enough".
The direct comparison test, for example, states that if $0\leq f(x)\leq g(x)$... | {
"language": "en",
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Some Matrix product $A \odot B$ I'm confronted with the following problem:
Let $G=(V,E)$ be a directed graph with edge costs $c:E\rightarrow \mathbb{R}$ (Negative cycles do not matter). Let $V=\{v_1,\dots,v_n\}$.
For Matrices $A$ and $B$ $\in \mathbb{R}^{n \times n}$, we define a matrix product $\odot$ as follows: $A \... | You can check by induction that the $(i,j)$th entry of $M^{\odot k}$ is the smallest weight of a path that (1) leads from $i$ to $j$, (2) contains $k$ edges.
As to the behavior of $M^{\odot k}$ as $k$ gets large.
(i) If the initial graph has a negative cycle, then we can move around it as many times as we want, so the ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How do I prove that finitely generated group with $g^2=1$ is finite? Let $G$ be a finitely generated group.
Assume for all $g\in G, g^2=e$.
Then, how do I show that $G$ is actually finite?
I don't know where to start..
| Let $\Bbb K=\Bbb Z/\Bbb Z$. And define $\Bbb K\times G\to G$ by $(k,g)\mapsto g^k$.
Hence $G$ is a $\Bbb K$ vector space with finite dimension . so $G$ is finite.
| {
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"timestamp": "2023-03-29T00:00:00",
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Deduce $ \forall x P(x) \vdash \exists xP(x) $ Well it's a little awkward but how can I show this in a natural deduction proof?
$ \forall x P(x) \vdash \exists xP(x) $
I think one has too proof that with a proof by contradiction rule but since I cannot eliminate the $ \exists $ quantifier I am stuck. I know this is a q... | For a proof with natural Deduction, we refer to Dirk van Dalen, Logic and Structure (5th ed - 2013) for the rules :
$$\frac{∀x \varphi(x) }{\varphi(t)} \text {∀E ; we require $t$ to be free for $x$ in $\varphi$ [page 86] }$$
$$\frac{\varphi[t/x] }{∃x \varphi} \text {∃I, with $t$ free for $x$ in $\varphi$ [page 93]}$$... | {
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Test for convergence: $\int_1^\infty \frac{\ln x}{x^2} \, dx$
Is this improper integral convergent or divergent?$$\int_1^\infty \frac{\ln x}{x^2} \, dx$$
I tried $\int_1^\infty \frac{\ln x}{x^2} \, dx \le \int_1^\infty \frac{\ln x}x \, dx$ but the RHS diverges, which makes this relation inconclusive. I think the inte... | First, we solve the integral using integration by parts (let me know if I should elaborate more on this)
$$\lim_{a\to\infty}\int_1^a \frac{\ln x}{x^2}\,dx=\lim_{a\to\infty}\left.\frac{-\ln x-1}{x}\right|_1^a=\Big(\lim_{a\to\infty}\frac{-\ln a-1}{a}\Big)+1$$
Now, we have to solve the limit
$$\Big(\lim_{a\to\infty}\frac... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Getting the cumulative distribution function for $\sqrt{X}$ from the cumulative distribution function for $X$ I've a data set $X$ which consists of randomly generated numbers.
My aim is to plot the cumulative distribution function for square root of $X$ without generating data set for square root of $X$. I'm using Math... | Hint:
$$ cdf_{\sqrt X}(x) = P(\sqrt X \le x) = P(X \le x^2) = cdf_X(x^2)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/977184",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Expectation of a Poisson Distribution: E[X(X-1)(X-2)(X-3)]
Given $X \sim Poi(\lambda)$, what is the expectation of $\mathbb{E}[X(X-1)(X-2)(X-3)]$?
I'm not sure how to approach this. I was thinking of expanding the polynomial, but that led to fairly ugly results. I was told that there is an elegant solution, but I can... | Hint:
$$e^{-\lambda}\sum_{n=0}^{\infty}n\left(n-1\right)\cdots(n-k+1)\frac{\lambda^{n}}{n!}=e^{-\lambda}\lambda^{k}\sum_{n=k}^{\infty}\frac{\lambda^{n-k}}{\left(n-k\right)!}=\cdots$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Limit of a 0/0 function Let's say we have a function, for example,
$$
f(x) = \frac{x-1}{x^2+2x-3},
$$
and we want to now what is
$$
\lim_{x \to 1} f(x).
$$
The result is $\frac{1}{4}$.
So there exists a limit as $x \to 1$.
My teacher says that the limit at $x=1$ doesn't exist. How is that? I don't understand it. We kn... | It's possible that your teacher was pointing out the fact that the function doesn't exist at $x = 1$. That's different from saying that the limit doesn't exist as $x \to 1$. Notice that by factoring,
$$
f(x) = \frac{x-1}{x^2 + 2x - 3} = \frac{x-1}{(x-1)(x+3)}
$$
As long as we are considering $x \ne 1$, the last expre... | {
"language": "en",
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Infinite Sum of Sines With Increasing Period A while ago, I was thinking about the Weierstrass function, which is a sum of sines with increasing frequencies in such a way that the curve is a fractal. However, I wondered what would happen if one took the sum where the frequencies decreased; in particular, noting that $|... | Another comment that is too long to be a comment:
The heuristic reason that the function is asymptotically proportional to $\sqrt{x}$ is that for very large $x$,
$\cdot$ The contributions of the terms in $S_n(x)$ for $n$ much less than $\sqrt{x}$ behave as pseudo-random numbers, restricted to $[-1,1]$. Thus $S_n(x)$ ... | {
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Is there an approximation to the natural log function at large values? At small values close to $x=1$, you can use taylor expansion for $\ln x$:
$$ \ln x = (x-1) - \frac{1}{2}(x-1)^2 + ....$$
Is there any valid expansion or approximation for large values (or at infinity)?
| x-1 is a crude approximation that is good for values of x close to 1.
2*(x-1)/(x+1) is a better approximation with a larger range.
6*(x-1)/(1 + 4 * Sqrt(x) + x) is better and has an even larger range.
90*(x-1)/(7+32 * x^(1/4) + 12 * x^(1/2) + 32 * x^(3/4) + 7 * x) is even better and has a larger range, but it never exc... | {
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Solve the integral $\int_0^\infty x/(x^3+1) dx$ I'm new here!
The problem: integrate from zero to infinity x over the quantity x cubed plus one dx. I checked on wolfram alpha and the answer is that the indefinite integral is this:
$$\int \frac{x}{1+x^3} dx = \frac{1}{6}\left(\log(x^2-x+1)-2 \log(x+1)+2 \sqrt{3} \arctan... | You should do something like this:
$$
\displaystyle\frac{x}{x^{3}+1} = \frac{A}{x+1}+\frac{Bx+C}{x^{2}-x+1}
$$
$$
\displaystyle \implies Ax^{2}-Ax+A+Bx^{2}+Bx+Cx+C = x
$$
$$
\implies (A+B)x^{2}+(-A+B+C)x+(A+C) = x
$$
$$
\implies A+B = 0
$$
$$
-A+B+C = 1
$$
$$
A+C = 0
$$
$$
\implies B = C = -A \implies -3A = 1 \implies ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/977692",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do you add two fractions? I have a fraction I am trying to solve. I know the answer already, as Wolfram says it is $\frac{143}{300}$.
The fraction is:
$$\frac{5}{12} + \frac{3}{50} = \space ?$$ Please explain why and how your method works.
| If $a=b$ then for any function $f(a)=f(b)$. Suppose
$\displaystyle x=\frac{5}{12}+\frac{3}{50}$. Then
$\displaystyle (12\cdot 50)\cdot x=(12\cdot 50)\cdot\left(\frac{5}{12}+\frac{3}{50}\right)$, so
$\displaystyle 600x=\frac{12\cdot 50\cdot 5}{12}+\frac{12\cdot 50\cdot 3 }{50}=$
$=50\cdot 5+12\cdot 3=286$, why $\disp... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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In what base does the equation $x^2 - 11x + 22 = 0$ have solutions $6$ and $3$? If we have below equation and know that $6$ and $3$ are answers of this equation, how to obtain the base used in the equation?
$$x^2 - 11x + 22 = 0$$
Partial result
The base is not $10$. (Because $3^2-3\cdot 11+22\ne 0$ in base $10$.)
| Knowing the roots you have
$$(x-3)(x-6) = x^2 -9x + 18$$
and therefore the base is 8. Check: $9=11_8$ and $18=22_8$.
| {
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Matrix A has eigenvalue $λ$ , Prove the eigenvalues of Matrix $(A+kI)$ is (λ + k)
The matrix A has an eigenvalue $λ$ with corresponding eigenvector $e$.
Prove that the matrix $(A + kI)$, where $k$ is a real constant and I is the identity matrix, has an eigenvalue $(λ + k)$
My Attempt:
$$(A + kI)e$$
$$= Ae + kIe = ... | The point is you need to find a non zero vector $v$ such that $(A+KI) v = \beta v$ and $ \beta $ is said to be the eigenvalue of $(A+KI)$.
So consider $ x \in \mathbb{R}^n$ such that $Ax=\lambda x$ then :
$(A+KI) x = Ax + Kx = \lambda x + Kx = (\lambda + K) x $ .
This implies that $\lambda+ K $ is an eigenvalue.
| {
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"timestamp": "2023-03-29T00:00:00",
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$2^{50} < 3^{32}$ using elementary number theory How would you prove; without big calculations that involve calculator, program or log table; or calculus that
$2^{50} < 3^{32}$
using elementary number theory only?
If it helps you: $2^{50} - 3^{32} = -727120282009217$, $3^{32} \approx$ $2^{50.718800023077\ldots}$, $3... | Compare:
$$3^{32}=(3^{2})^{16}\quad\text{vs.}\quad2^{50}=4(2^{3})^{16}$$
So that using the binomial theorem to second order:
$$\frac{3^{32}}{2^{50}}=\frac{(9/8)^{16}}{4}=\frac{(1+1/8)^{16}}{4}
>\frac{1+16/8+120/64}{4}>1$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Integrate $\int\frac{dx}{(x^2+1)\sqrt{x^2+2}}$ I would like some guidance regarding the following integral:
$$\int\frac{dx}{(x^2+1)\sqrt{x^2+2}}$$
EDIT: The upper problem was derived from the following integral $$\int\frac{\sqrt{x^2+2}}{x^2+1}dx$$
Where I rationalized the numerator which followed into: $$\int\frac{dx}{... | There is a general formula for it. $$\int \frac{dx}{(x^2+1)\sqrt{x^2+a}}=\frac{1}{\sqrt{a-1}}\tan^{-1}\left(\frac{\sqrt{a-1}x}{\sqrt{x^2+a}}\right)+C\tag{1}$$
$a=2$ gives $$\int \frac{dx}{(x^2+1)\sqrt{x^2+2}}=\tan^{-1}\left(\frac{x}{\sqrt{x^2+2}}\right)+C$$
Formula $(1)$ can be proven by substitution : $t=1/x$, $s=\sq... | {
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Show $\sum_n \frac{z^{2^n}}{1-z^{2^{n+1}}} = \frac{z}{1-z}$ Show $\displaystyle\sum_{n=0}^\infty \frac{z^{2^n}}{1-z^{2^{n+1}}} = \frac{z}{1-z}$ for $|z|<1$.
This is an additional problem for my complex analysis class and I've attempted it for a few hours but ended up taking wrong routes. All of my attempts I haven't us... | The first term is $z/(1-z^2)$ which is a series where every exponent is odd.
What are the exponents in the series of the second term?
| {
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"url": "https://math.stackexchange.com/questions/978266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
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Can $\mathbb{R}$ be partitioned into $n$ dense sets with same cardinality? Are there sets $S_i\subseteq\mathbb{R}$ with $i\leq n$ such that
*
*$S_i$ are disjoint,
*$S_i$ have same cardinality,
*$S_i$ are dense in $\mathbb{R}$?
| Partition the set of cosets of $\mathbb{Q}$ into $n$ sets of equal uncountable cardinality, and take the union of each partition element. The resulting partition of $\mathbb{R}$ consists of $n$ sets, each contains a coset of $\mathbb{Q}$ and so is dense, and their cardinalities are equal.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/978355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Estimating the behavior for large $n$ I want to find how these coefficients increase/decrease as $n$ increases:
$$ C_n = \frac{1}{n!} \left[(n+\alpha)^{n-\alpha-\frac{1}{2}}\right]$$
with $\alpha=\frac{1}{br-1}$ and $0\leq b,r \leq 1$.
I used the Stirling's Approximation factorial $n!\sim \sqrt{2\pi n} n^n e^{-n}$ a... | Taking the reciprocal of Stirling's Asymptotic expansion as derived in this answer:
$$
n!=\frac{n^n}{e^n}\sqrt{2\pi n}\left(1+\frac{1}{12n}+\frac{1}{288n^2}-\frac{139}{51840n^3}+O\left(\frac{1}{n^4}\right)\right)
$$
we get
$$
\frac1{n!}=\frac{e^n}{n^n}\frac1{\sqrt{2\pi n}}\left(1-\frac{1}{12n}+\frac{1}{288n^2}+\frac{13... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/978422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Countable Union to Countable Disjoint Union In many texts, the construction of a countable disjoint union of sets from a sequence of sets, $E_1, E_2,E_3,\ldots$ follows from:
Let $F_1 = E_1, F_2 = E_2\setminus E_1,F_3 = E_3\setminus (E_1\cup E_2),\ldots,F_n=E_n \setminus \bigcup\limits_{k=1}^{n-1} E_k$, etc.
I'm wonder... | Clearly, $F_1=E_1$ and $F_n=E_n\setminus E_{n-1}$, for $n>1$, implies that
$$
\bigcup_{n\in\mathbb N} F_n\subset\bigcup_{n\in\mathbb N} E_n. \tag{1}
$$
To show the opposite direction, let $x\in \bigcup_{n\in\mathbb N} E_n$. Then $x\in E_n$,
for some $n\in\mathbb N$. Find all such $n$'s for which $x\in E_n$, and pick th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/978519",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
"Introducing something extra" In my textbook "Calculus, Concepts and Contexts" - by James Stewart
There is a section on problem solving methods called "Introducing something extra"
Given the problem:
$$ \lim_{x \to 0} \frac{ \sqrt[3]{1+cx}-1}{x}$$ where $c$ is constant - Stewart suggests this method and introduces a n... | Introducing a "new variable" amounts to saying that
$$
\lim_{x\to a} f(x) = \lim_{y \to b} f(g(y))
$$
when
*
*$\lim_{y \to b} g(y) = a$
*$g$ is nonconstant near $b$
*The right hand limit exists.
This is an oft-used but seldom proven theorem. It amounts to the proof that the limit of the composition is the comp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/978624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How do I make sense of $\{ n \} \cap n$? I've been learning set theory, and I've come across an exercise in which I'm trying to prove that $\forall x \forall y x \in y \rightarrow y \neq x$. I want to use the axiom of foundation to prove this, but I'm stuck making sense of that axiom for the base case in which a set co... | In axiomatic set theory, everything is a set. In particular the integers. Which sets are the integers may vary, but the standard is to use von Neumann's definition:
*
*$0=\varnothing$, and
*$n+1=n\cup\{n\}$.
So for example $3=\{\varnothing,\{\varnothing\},\{\varnothing,\{\varnothing\}\}=\{0,1,2\}$.
And you don't... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/978726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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For every integer $n \geq 1$, prove that $3^n \geq n^2$. It's been a while since I've done induction, and I feel like I'm missing something really simple. What I have is this:
Base Case: $n=1$
$$3^n \geq n^2 \implies 3 \geq 1$$
Inductive Hypothesis
For all integers $1 \leq n < n+1$:
$$3^n \ge n^2$$
Inductive Ste... | For $n>1$ we have $2n(n-1)\geq 1$, thus:
$3^n\geq n^2\rightarrow 3^{n+1}\geq 3n^2=n^2+2n^2$
According to the first line of the answer, $2n(n-1)\geq 1\rightarrow 2n^2\geq 2n+1$
So we can say:
$3^{n+1}\geq n^2+2n+1=(n+1)^2$
Hence the assumption is proved by induction
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/978828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Number of distinct real roots of $x^9 + x^7 + x^5 + x^3 + x + 1$ The number of distinct real roots of this equation $$x^9 + x^7 + x^5 + x^3 + x + 1 =0$$ is
Descarte rule of signs doesnt seems to work here as answer is not consistent . in general i would like to know nhow to find number of real roots of any degree equ... | Denote the polynomial as $p$.
$p$ have at least one real root because it is of odd degree (this follows from IVT).
Assume $p$ have more then one real root, then there are two different points $x_1,x_2$ s.t $p(x_1)=p(x_2)=0$
By Rolle's theorem this imply there is $x_1<c<x_2$ s.t $p'(x)=0$ - show such a point does not ex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/978918",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
derivative problem. is it same? First derivative of $y=\ln(x)^{\cos x}$ is $-\sin x\ln x+\frac{\cos x}{x}$ or another answer? My friend gets another answer, but it's true? thanks.
| Assuming that $\ln (x)^{\cos x}=\ln\left(x^{\cos x}\right)$, we have
$$ \frac{d}{dx}\left[\ln x^{\cos x}\right]=\frac{d}{dx}\left[(\cos x)\ln x\right]=(\cos x)\frac{d}{dx}\left[\ln x\right]+(\ln x)\frac{d}{dx}\left[\cos x\right] $$
$$ =(\cos x)\frac{1}{x}+(\ln x)(-\sin x)=\frac{\cos x}{x}-(\sin x)\ln x $$
If you meant ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/978996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
The reflection of $f(x,y) = x^2 - y^2$ How would I make a reflection of $$ f(x,y) = x^2 - y^2 $$ along the z axis? Beacuse if if write $$ f(x,y) = -(x^2 - y^2) $$, flips the figure along the XY axis...
| Reflection in the $z$-axis takes $(a,b,c)$ to $(-a,-b,c)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/979088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Does $1.0000000000\cdots 1$ with an infinite number of $0$ in it exist? Does $1.0000000000\cdots 1$ (with an infinite number of $0$ in it) exist?
| In the hyperreal number system which is an extension of the real number system you have infinite integers and the corresponding extended decimal expansions where it is meaningful to talk about digits at infinite rank (more precisely, rank defined by an infinite integer). In this system your decimal makes sense.
Extend... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/979177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "40",
"answer_count": 8,
"answer_id": 3
} |
If $(a + ib)^3 = 8$, prove that $a^2 + b^2 = 4$
If $(a + ib)^3 = 8$, prove that $a^2 + b^2 = 4$
Hint: solve for $b^2$ in terms of $a^2$ and then solve for $a$
I've attempted the question but I don't think I've done it correctly:
$$
\begin{align*}
b^2 &= 4 - a^2\\
b &= \sqrt{4-a^2}
\end{align*}
$$
Therefore,
$$
\... | Let $z=a+ib,$ then it is given that $z^3=8.$ Therefore taking the modulus of both sides $|z|^3=8.$ Hence $|z|=2$ and $|z|^2=a^2+b^2=4.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/979252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 7,
"answer_id": 1
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Question about intersection/union of a set and its complement I was answering this multiple choice question from this website examtimequiz.com/maths-mcq-on-sets:
If $A$ is any set, then
*
*$A \cup A' = U$
*None of these
*$A \cap A' = U$
*$A \cup A' = \emptyset$
I answered (1), but apparently the correct answer i... | I am assuming that $A'$ = the complement of set $A$, and that $U$ = the universal set. The red part is the set $A$, and $A'$ is everything in the white. The set $U$ is everything, so the red part + the white part. That makes $A \cup A' = U$. So there was a mistake in the question and your answer is correct.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/979368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Proving ring $R$ with unity is commutative if $(xy)^2 = x^2y^2$
Let $R$ be a ring with unity. Assume that $\left(xy\right)^2 = x^2 y^2$ for all $x, y \in R$. Prove that $R$ is commutative.
I tried several methods to solve this but couldn't get through. Now the solution in almost all the textbooks goes like this.
Firs... | By Dan Shved's comments, you can see why switching $\Box$ by $\Box +1$ is a commen way in commutativity conditions. The commutator $[x,y]$ plays an vital role in commutativity conditions. There is immense number of papers which are about this technique. For example:
1) I. N. HERSTEIN, Two remarks on the commutativity o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/979474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Prove or disprove if $G$ is 2-edge-connected then there exist 2 edges disjoint $u-v$ trail such that every edge of $G$ lies on one of these trails. Let $G$ be a connected graph with exactly 2 odd vertices $u$ and $v$ such that $deg(u) \geq 3$ and $deg(v) \geq 3$. Prove or disprove if $G$ is 2-edge-connected then there ... | If a connected graph has exactly two odd vertices it has an eulerian trail between them. Why? Call those vertices $uv$. If $uv$ is in the graph remove it, if it is not in the graph add it. You now have a graph where all the vertices have even degree, thus an eulerian circuit, that circuit becomes a path when you take t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/979592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Largest $n$-vertex polyhedron that fits into a unit sphere In two dimensions, it is not hard to see that the $n$-vertex polygon of maximum area that fits into a unit circle is the regular $n$-gon whose vertices lie on the circle: For any other vertex configuration, it is always possible to shift a point in a way that i... | This is supposed to be a comment but I would like to post a picture.
For any $m \ge 3$, we can put $m+2$ vertices on the unit sphere
$$( 0, 0, \pm 1) \quad\text{ and }\quad \left( \cos\frac{2\pi k}{m}, \sin\frac{2\pi k}{m}, 0 \right) \quad\text{ for }\quad 0 \le k < m$$
Their convex hull will be a $m$-gonal bipyramid w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/979660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 1,
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Prove that $\mathcal B(\mathbb R)\times \mathcal B(\mathbb R)\subseteq \mathcal B (\mathbb R^2)$ I need to prove that
$$\mathcal A(\mathcal B(\mathbb R)\times \mathcal B(\mathbb R))= \mathcal B (\mathbb R^2)$$
Where $\mathcal B$ is the generated Borel algebra and $\mathcal A$ is the generated $\sigma$-algebra. I've red... | Hints:
*
*Show that $$\mathcal{D} := \{A \in \mathcal{B}(\mathbb{R}); \forall O \subseteq \mathbb{R} \, \text{open}: A \times O \in \mathcal{B}(\mathbb{R}^2)\}$$ is a $\sigma$-algebra and conclude that $\mathcal{B}(\mathbb{R}) \times \mathcal{O} \subseteq \mathcal{B}(\mathbb{R}^2)$. ($\mathcal{O}$ denote the open se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/979714",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Proof a number is Fibonacci number I have a question regarding the proof that a number n is a Fibonacci number if and only if $5n^2-4$ or $5n^2+4$ is a perfect square. I don't understand the second part of the proof: knowing that $5n^2-4$ or $5n^2+4$ is a perfect square, prove that n is a Fibonacci number.
I attach th... | The ring of integers $\mathcal{O}_K$ in the quadratic number field $K=\mathbb{Q}(\sqrt{5})$ is given by $\mathbb{Z}+\frac{1+\sqrt{5}}{2}\mathbb{Z}$. This is a basic result of algebraic number theory. So the author means that these elements are in the ring of integers of the extension field. Furthermore the ring $\mathc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/979804",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Is abstract algebra (mostly?) restricted to $2$-ary operators? This may be due to my own pure ignorance but it's my experience that all abstract algebra I've been introduced to, both in actual courses and in self-studies only exclusively deals with algebraic objects consisting of a set together with one or more binary ... | In logic the operators are 2-ary because any function $\mathbb Z_2^n\rightarrow \mathbb Z_2$ can be expressed by 2-ary operators. In mathematics it's only because that abstract algebra is a generalization of the numbers and their common operators.
To study general n-aries in the same manner would require a lot of new e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/979916",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
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Show that $\frac{1}{e^\gamma \text{log }x + O(1)} = \frac{1}{e^\gamma\text{log }x} + O\left(\frac{1}{(\text{log }x)^2}\right)$ Show that $\frac{1}{e^\gamma \text{log }x + O(1)} = \frac{1}{e^\gamma\text{log }x} + O\left(\frac{1}{(\text{log }x)^2}\right)$
I'm using one of Merten's estimates in a proof, the one that state... | Would an answer like this work?:
\begin{align}
\frac{1}{e^\gamma\text{log }x + O(1)} &= \frac{1+O(\text{log }x)}{e^\gamma\text{log }x + O((\text{log }x)^2)}\\
&=\frac{1}{e^\gamma\text{log }x}\left(\frac{1+O(\text{log }x)}{1+O(\text{log }x)}\right)\\
&=\frac{1}{e^\gamma\text{log }x}(1+O(\text{log }x))
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/980123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Understanding homomorphism and kernels Let $G$ be a group and $\phi$ a Homomorphism
$$
\phi:G\to G'
$$
Now I know that the size of the kernel tells you how many elements in $G$ map to the same element in $G'$
I couldn't find this in my book, but I have concluded the following.
$$
\frac{|G|}{| \:\text{ker} \: \phi \:|}... | This is close, but not quite right. For example, if $G^\prime$ is a proper subgroup of $H$ then $\phi$ defines a homomorphism
$$
\bar\phi:G\to H
$$
with $\ker\bar\phi=\ker\phi$. Your result would then imply
$$
\lvert G^\prime\rvert=\frac{\lvert G\rvert}{\lvert\ker \phi\rvert}=\frac{\lvert G\rvert}{\lvert\ker\bar\phi\rv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/980264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
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Combinatorics question. Probability of poker hand with one pair If we assume that all poker hands are equally likely, what is the probability of getting 1 pair?
So the solution is
I understand numerator part, but I do not understand why in denominator we have 3!.
In the books it says "The selection of the cards “b”,... | We count the one pair hands in a somewhat different way. The kind we have a pair in can be chosen in $\binom{13}{1}$ ways. For every choice of kind, the actual cards can be chosen in $\binom{4}{2}$ ways.
Now we choose the $3$ kinds that we have singletons in. These can be chosen in $\binom{12}{3}$ ways. Imagine arrangi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/980337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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The greatest common divisor is the smallest positive linear combination How to prove the following theorems about gcd?
Theorem 1: Let $a$ and $b$ be nonzero integers. Then the smallest positive linear combination of $a$ and $b$ is a common divisor of $a$ and $b$.
Theorem 2: Let $a$ and $b$ be nonzero integers. The gcd... | The procedure very briefly sketched in your comment is the standard way to prove Theorem 1.
For Theorem 2, the proof depends on exactly how the gcd of $a$ and $b$ is defined. Suppose it is defined in the naive way as the largest number which is a common divisor of $a$ and $b$.
We then need to show that there cannot b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/980426",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How should Theorem 3.22 in Baby Rudin be modified so as to yield Theorem 3.23 as a special case? I'm reading Walter Rudin's PRINCIPLES OF MATHEMATICAL ANALYSIS, 3rd edition, and am at Theorem 3.22.
Theorem 3.22: Let $\{a_n\}$ be a sequence of complex numbers. Then $\sum a_n$ converges if and only if for every $\epsilo... | *
*In this kind of proof "something $\leq \epsilon$" it does not matter "$\leq$" or "$<$", since it carries something like "for any sufficiently small $\epsilon$".
*The double implication is lost since $m$ can be much larger than $n$, and if "$a_n$ does not decreases fast enough", you cannot control $\sum_{k=n}^m a_k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/980474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to decide about the convergence of $\sum(n\log n\log\log n)^{-1}$? In Baby Rudin, Theorem 3.27 on page 61 reads the following:
Suppose $a_1 \geq a_2 \geq a_3 \geq \cdots \geq 0$. Then the seires $\sum_{n=1}^\infty a_n$ converges if and only if the series
$$ \sum_{k=0}^\infty 2^k a_{2^k} = a_1 + 2a_2 + 4a_4 + 8... | Using this method which's called Cauchy condensation we get
$$\sum_{k\ge1}\frac{2^k}{2^k\ln 2^k\ln\ln(2^k)}=\frac1{\ln2}\sum_{k\ge1}\frac1{k\ln(k\ln2)}\sim\frac1{\ln2}\sum_{k\ge1}\frac1{k\ln(k)}$$
so the series
$$\sum_{n\ge1}\frac1{n\ln n\ln(n\ln n)}$$
is divergent. Can you now solve the second series?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/980573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 1
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is the lexicographic order topology on the unit square connected/path connected? I was wondering, given the lexicographic order topology on $S=[0,1] \times [0,1]$, is it connected (and path connected)?
I found a reference to Steen's and Seebach's Counterexamples in Topology, and in page 73 they say that:
Since in th... | Suppose $X$ ($=S$?) is path connected, so there exists a continuous path $\gamma : [a,b] \to X$ such that $\gamma(a)=(0,0)=P$ and $\gamma(b)=(1,1)=Q$. Since every point of $X - \{P,Q\}$ disconnects $P$ from $Q$ it follows that $\gamma$ is surjective. For each $t \in [0,1]$ let $J_t$ be the open vertical segment with lo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/980647",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
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Does every plane curve contain a rational point? Does every plane curve contain a rational point?
I think the answer is yes, but I can not prove this. Please help.
However, if it is possible to build a pathological curve - without rational points, then even more interesting question arises - which properties of a curve... | The answer is NO. We can show more by only considering straight lines and the fact that rational points are countable.
Choose an arbitrary point $A$ in ${{\mathbb{R}}^{2}}$ whose coordinates are both irrational. The set $L = \left\{ l:A\in l \right\}$ is uncountable, and thus there is no one-to-one mapping from $L$ to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/980741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Proof of a summation of $k^4$ I am trying to prove $$\sum_{k=1}^n k^4$$
I am supposed to use the method where $$(n+1)^5 = \sum_{k=1}^n(k+1)^5 - \sum_{k=1}^nk^5$$
So I have done that and and after reindexing and a little algebra, I get $$(n+1)^5 = 1+ 5\sum_{k=1}^nk^4 + 10\sum_{k=1}^nk^3 + 10\sum_{k=1}^nk^2 + 5\sum_{k=1}... | Here's an alternative approach using binomial coefficients.
Firs we express $k^4$ as a linear combination of $\binom{k+a}{4}$ where $a=0,1,2,3$, i.e.
$$k^4={k+3\choose 4}+11{k+2\choose 4}+11{k+1\choose 4}+{k\choose 4}$$.
Summing this from $1$ to $n$ and using the hockey stick summation identity, i.e. $\sum_{r=0}^m {r\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/980843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Generation function for recurrence Could you tell me how I can find the generation function for recurrence $\sum_{n = 0}^{ \infty} n a_n t^n$ if I know $A(t)$ - generation function for $a_0, a_1, a_2 \dots$ .
Thanks
| You know that:
$$\sum_{n=0}^{\infty}a_nt^n=A(t)$$
Derive both sides (left side term by term).You get:
$$\sum_{n=0}^{\infty}na_nt^{n-1}=A'(t)$$
Now multiply both sides by $t$:
$$\sum_{n=0}^{\infty}na_nt^{n}=tA'(t)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/980915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Polynomial Arithmetic Modulo 2 (CRC Error Correcting Codes) I'm trying to understand how to calculate CRC (Cyclic Redundancy Codes) of a message using polynomial division modulo 2. The textbook Computer Networks: A Systems Approach gives the following rules for division:
*
*Any polynomial $B(x)$ can be divided by a ... | Your calculations are correct.
It is worth keeping in mind that the quotient is not of importance in the CRC calculations, it is the remainder that is needed. The careful calculations
that you have carried out and written up in detail are good for familiarizing
oneself with the algorithm. However, after some practice, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/980989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Can a relation be a partial order and an equivalence at the same time? Can a relation be a partial order AND an equivalence at the same time?
For instance, if we have a set $A = \{1, 2, 3, 4, 5\}$ and a relation $R$ on $A$ defined as $R = \{(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)\}$: this relation is reflexive, anti-sym... | Two remarks:
(1) Equality is an equivalence relation which is also a partial order ($\leq$).
(2) An equivalence relation is never a strict partial order ($<$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/981094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Does $\int_{3}^{\infty}\frac{1}{(x-2)^{3/2}}\text{ d}x$ converge? I am trying to see whether or not
$$\int\limits_{3}^{\infty}\dfrac{1}{(x-2)^{3/2}}\text{ d}x$$
converges. My first instinct was to notice in $[3, \infty)$ that
$$\dfrac{1}{(x-2)^{3/2}} > \dfrac{1}{x^{3/2}}\text{.} $$
But $\displaystyle\int\limits_{3}^{... | $$\int\limits_3^\infty\frac1{(x-2)^{3/2}}dx=\left.-2(x-2)^{-1/2}\right|_3^\infty:=-2\lim_{b\to\infty}\left(\frac1{\sqrt{b-2}}-1\right)=2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/981184",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
If two harmonic quartets have a common point, prove their lines are concurrent Let $A,B,C,D$ and $A,L,M,N$ be collinear points such that $\{AB,CD\} = \{AL,MN\} = -1$. Prove that the lines BL, CN and DM concur.
I tried to build a triangle using A as a common point and then use Ceva Theorem, but the concurrent lines aren... | A perspectivity, i.e. a projection from one line onto another through a common center, preserves cross ratios.
Consider the point where $BL$ and $CM$ intersect. Since the lines intersect in $A$, this perspectivity will map $A$ to itself. So you have three points $A,B,C$ and their images $A,L,M$. Therefore the image of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/981261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Calculus 1: Find the limit as x approaches 4 of $\frac{3-\sqrt{x+5}}{x-4}$ I understand how to find limits, but for some reason I cannot figure out the algebra of this problem. I tried multiplying by the conjugate and end up with 0/0. When I check on my calculator, or apply L'Hopital's rule I get -1/6. Is there an alge... | $$\begin{array}{rcl}\lim_{x\to 4} \frac{3-\sqrt{x+5}}{x-4} & = & \lim_{x\to 4} \frac{(3-\sqrt{x+5})(3+\sqrt{x+5})}{(x-4)(3+\sqrt{x+5})}=\lim_{x\to 4} \frac{9-(x+5)}{(x-4)(3+\sqrt{x+5})} \\ & = & \lim_{x\to 4} \frac{4-x}{(x-4)(3+\sqrt{x+5})} =\lim_{x\to 4} \frac{-1}{3+\sqrt{x+5}}=- \frac{1}{6}.\end{array}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/981332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Variance & Joint Density Function $X$ and $Y$ have joint density given by $$f_{XY}(x,y)=\begin{cases}2,& 0≤x≤y≤1 \\0,& \text{elsewhere}\end{cases}$$ a) Find $\text{Var}(Y|X=x_0)$.
b) What is the answer if $x_0$ is not in the interval $[0,1]$?
So I know that if the $x_0$ is not in the interval, then the answer is $0$,... | For $0\le x \le y \le 1$ you have that $$f_{Y|X}(y|x)=\dfrac{f_{XY}(x,y)}{f_X(x)}=\frac{2}{\int_{x}^{1}f_{XY}(x,y)dy}=\dfrac{2}{\int_{x}^{1}2dy}=\dfrac{1}{1-x}$$ for all $x\le y \le 1$. That is $Y|X=x$ is uniformly distributed in $[x,1]$. Thus $$Var(Y|X=x)=\frac{(1-x)^2}{12}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/981604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Distance function is in fact a metric I know I should be able to show this, but for some reason I am having trouble. I need to show that $$d(x,y) = \frac{|x-y|}{1+|x-y|}$$ is a metric on $\Bbb R$ where $|*|$ is the absolute value metric. I am getting confused trying to show that the triangle inequality holds for this f... | Hint:
Put $f(t) = \frac{t}{1+t} $. Verify yourself that
$$ f'(t) = \frac{1}{(1+t)^2 } $$
Hence, $f'(t) \geq 0 $ for all $t$. In particular $f$ is an increasing function. In other words, we have
$$ |x+y| \leq |x| + |y| \implies f(|x+y|) \leq f(|x|+|y|) \implies .....$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/981684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Regular Octagon Area Doing some maths homework I came across the area of a regular octagon on Google. This was given by:
$$ A=2(1+\sqrt{2})a^2 $$
I thought this looked rather ugly and slightly complicated and so began to look at regular octagons myself (Yes, I'm a nerd :)!). I managed to re-write the equation to
$$ A=\... | Assuming that $a$ and $x$ are both the length of one side of the regular octagon, the results are the same:
$$\frac{1}{\sqrt{2} \sin^2 (\pi/8)} = \frac{1}{\sqrt{2}} \left(\frac{2}{1 - \cos(\pi/4)}\right) = \frac{\sqrt{2}}{1 - \frac{1}{\sqrt{2}}} = 2(1 + \sqrt{2}).$$
The area formula for the general regular $n$-gon just... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/981754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Temperature defined on a tetrahedron I am asked to prove that the temperature of a tetrahedron must have at least three distinct points on the edges or vertices of the tetrahedron with the same value. I may assume that the temperature is a continuous function.
Is the following reasoning correct?
Consider two vertices ... | Your argument implicitly assumes that $T_a\ne T_b$. (In fact, your notation $(T_a,T_b)$ also suggests that $T_a<T_b$.) This need not be the case. To complete the proof, consider two cases:
*
*All vertices have the same temperature. [conclusion is immediate]
*There are two vertices of unequal temperature: label th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/981863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Is the path between 2 vertices of a Minimum weight tree of a graph the shortest path between those 2 vertices? Suppose we have an undirected, connected graph, $G_1$
If you have a minimum weight spanning tree $G_2$ for graph $G_1$. Is it possible to find two vertices in $G_1$ which is has a shortest path that is not pre... | HINT:
Think of a triangle $ABC$ with $AB=3$, $BC=4$, $AC=5$. The minimum spanning tree contains the edges $AB$, $BC$ and $AB+ BC > AC$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/981947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
a question about contractions on Hilbert spaces Let $\cal{H}$ be a Hilbert space, $T_1,T_2\in\cal{B(H)}$,
*
*$\|T_1(h_1)+T_2(h_2)\|^2\leq\|h_1\|^2+\|h_2\|^2$ for all $h_1,h_2\in\cal{H}$.
*$T_1T^\ast_1+T_2T^\ast_2\leq I$.
Then can we verify that 1 holds if and only if 2 holds ?
| Consider the operator $T:\mathcal H\oplus \mathcal H\to\mathcal H$ defined by
$$T(h_1\oplus h_2):= T_1(h_1)+T_2(h_2)\, . $$
Then (1) says exactly that $\Vert T\Vert\leq 1$. Since $\Vert T\Vert=\Vert T^*\Vert$, this is equivalent to the condition $\Vert T^*\Vert\leq 1$, which is again equivalent to
$$TT^*\leq I\, .$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/982143",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
A function $f(x)$ that Riemann integrable on $[a,b]$. Define a function $f(x)$ that Riemann integrable on $[a,b]$.
Let
$$g(x)=\begin{cases}
f(x)&\text{if}&x\in[a,b], \\
f(a)&\text{if}&x<a, \\
f(b)&\text{if}&x>b.
\end{cases}$$
Let $\delta >0$,define$$F_{\delta}(x)=\frac{1}{\delta}\int_{0}^{\delta} (g(x+t)-g(... | I suppose that $0<\delta<b-a$.
Your last equality
$$\int_{0}^{\delta}\left(\int_{a}^{b}( g(x+t)-g(x))dx\right)dt=\int_{a}^{b}\left(\int_{0}^{\delta} (g(x+t)-g(x))dt\right)dx$$
is true by Fubini's theorem.
Now we have:
$$\int_{a}^{b}( g(x+t)-g(x))dx=\int_{a+t}^{b+t}g(u)du-\int_a^b g(x)dx=\int_b^{b+t}g(u)du-\int_a^{a+t}g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/982231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Area of square created by intersection of segments from a square vertexes to their opposite sides There will be an square created when we draw segments from a square vertexes to their opposite sides' middle.
What is the relation between smaller square's area and the side length of the bigger one?
| Let $a$ be the side length of the grey square (and $1$ the side lengthg of the original square).
By similarity, the length of the line segment from $D$ to $DE\cap CH$ is also $a$.
Then the triangle with base $AE$ complete the quadrilateral with top edge $DE$ to a square of area $a^2$. We can do the same with the other ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/982376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove $OD$ is the angle bisector of the angle BOC Let $ABC$ be a non-isosceles triangle and $I$ be the intersection of the three internal angle bisectors. Let $D$ be a point of BC such that $ID\perp BC$ and $O$ be a point on AD such that $IO\perp A$D . Prove $OD$ is the angle bisector of the angle BOC.
| This is complicated and I have to break it into 3 parts.
Fact#1) In the figure, $BOU$ is a straight line and is composed of angles at $O$ with $\alpha + \beta + \phi + \theta = 180^0$. It is given that $\angle IOA = \beta + \phi = 90^0$. If $\alpha = \beta$, then $\phi = \theta$. This is obvious and therefore the proo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/982491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
prove that this equality is always right for each positive x and y. prove that this inequality is hold for each positive x,y.
$x\over\sqrt{y}$ + $y\over\sqrt{x}$ $\ge$ $\sqrt{x}$ + $\sqrt{y}$
I want a detailed way of solving the question.
| Without loss of generality you can assume $0<x<y$. Then use the rearrangement inequality:
$$
\sqrt x + \sqrt y = \frac{x}{\sqrt x} + \frac{y}{\sqrt y} \le
\frac{x}{\sqrt y} + \frac{y}{\sqrt x}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/982701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Find the value of $\,\, \lim_{n \to \infty}\Big(\!\big(1+\frac{1}{n}\big)\big(1+\frac{2}{n}\big) \cdots\big(1+\frac{n}{n}\big)\!\Big)^{\!1/n} $ What is the limit of:
$$
\lim_{n \to \infty}\bigg(\Big(1+\frac{1}{n}\Big)\Big(1+\frac{2}{n}\Big)
\cdots\Big(1+\frac{n}{n}\Big)\bigg)^{1/n}?
$$
By computer, I guess the limit is... | Let $f(n)=[\prod_{i=0}^n (1+r/n)]^{1/n}$ Then, $\ln f(n)=1/n\sum_{r=0}^n\ln (1+r/n)\Rightarrow \lim_{n\to \infty}\ln f(n)=\int_{0}^1 \ln (x+1) dx=\ln 2-\int_{0}^1 \dfrac{x}{x+1}dx=2\ln 2-1\Rightarrow \lim_{n\to \infty} f(n) =4/e$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/982772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
solve indefinite integral I have this indefinite integral $\int 3 \sqrt{x}\,dx$ to solve.
My attempt:
$$\int 3 \sqrt{x}\,dx = 3 \cdot \frac {x^{\frac {1}{2} + \frac {2}{2}}}{\frac {1}{2} + \frac {2}{2}}$$
$$\int 3 \sqrt{x}\,dx = 3 \frac{x^{\frac {3}{2}}}{\frac {3}{2}} = \frac{2}{3} \cdot \frac{9}{3} x^{\frac {3}{2}}$$
... | In your penultimate step, $\frac 23$.$\frac 93$ $=\frac {18}{9}$ $=2$, not 6.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/982826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Number of $K_{10}$ always increases Let $G=(V,E)$ be a graph with $n\geq 10 $ vertices. Suppose that when we add any edge to $G$, the number of complete graphs $K_{10}$ in $G$ increases. Show that $|E|\geq 8n-36$.
[Source: The probabilistic method, Alon and Spencer 3rd ed., p.12, problem 5]
For the base case $n=10$, we... | The solution to this problem uses a clever application of Theorem 1.3.3 (in "The Probabilistic Method, 3rd edition"). Since not everyone has access to the text, I will state the theorem (with necessary definitions) here first.
Definition: Let $\mathcal{F}=\{(A_{i}, B_{i})\}_{i=1}^{h}$ be a family of pairs of subsets o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/982902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 2,
"answer_id": 0
} |
The value of $\int_0^{2\pi}\cos^{2n}(x)$ and its limit as $n\to\infty$
Calculate $I_{n}=\int\limits_{0}^{2\pi} \cos^{2n}(x)\,{\rm d}x$
and show that $\lim_{n\rightarrow \infty} I_{n}=0$
Should I separate $\cos^{2n}$ or I should try express it in Fourier series?
| Here is a completely
elementary
(i.e., nothing beyond basic integration)
proof.
Taking advantage of
the symmetries of
$\cos$,
$I_{n}
=\int\limits_{0}^{2\pi} \cos^{2n}(x)\,{\rm d}x
=4\int\limits_{0}^{\pi/2} \cos^{2n}(x)\,{\rm d}x
=4\int\limits_{0}^{\pi/2} (\cos^2(x))^{n}\,{\rm d}x
=4\int\limits_{0}^{\pi/2} (1-\sin^2(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/983017",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
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