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32 votes
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Why do I get two different answers while doing this simple limit math?

In attempt 2, your method uses the fact that if you have two convergent sequences $a_n\to a<\infty$, $b_n\to b<\infty$, then the product is also convergent: $a_nb_n\to ab<\infty$. The issue ...
S.L.'s user avatar
  • 1,088
29 votes

What is the mean value of $|\sin x +\sin (\pi x)|$?

This is far, FAR from a rigorous answer to your question, but I think the heuristic behind it is interesting. Even more interesting is that I reached the exact same answer you did numerically. ...
Willow Wisp's user avatar
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29 votes
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Asymptotic formula for ratio of double factorials

If you take an extra term in Stirling's approximation, you have $n! = \sqrt{2\pi n}\left(\frac{n}{e}\right)^n \left(1 +\frac{1}{12n} + O(\frac1{n^2})\right)$ $(2n)! =\sqrt{4\pi n}\left(\frac{2n}{e}\...
Henry's user avatar
  • 156k
25 votes

Possible error in Stanley's combinatorics volume 1

This is problem 10 in Exercises for Chapter 2 in Volume 1, 2nd Ed. on page 222 of Enumerative Combinatorics by R. P. Stanley. In the Errata sheet the author states: page 222, Exercise 10(b), lines 1 ...
Markus Scheuer's user avatar
21 votes

In an infinite sum, is there an actual term at an infinite position?

You ask, among other things, if $n$ has to equal infinity. In fact, $n$ cannot equal infinity. What $n$ does do is take on more and more values—every one of which is finite—without end. Infinity is ...
Paul Tanenbaum's user avatar
21 votes
Accepted

Trying to calculate a limit with a finite product and WolframAlpha disagrees with my logic.

You're right and WolframAlpha is wrong. Note that the first factor is $\frac2{N+2}$ and all the other factors (as you noted) are less than $1$. So the product is actually less than $\frac2{N+2}$, ...
Greg Martin's user avatar
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21 votes
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Is there any sequence of closed shapes whose limit tends to the unit circle while the limit of the perimeter goes to infinity?

In the annulus with inner radius $1-1/n$ and outer radius $1+1/n$ draw a curve that oscillates back and forth between the inner and outer edge often enough to have length $n$. Then the limit as $n \to ...
Ethan Bolker's user avatar
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20 votes
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What is $\lim\limits_{n\to\infty}\frac1n \left(\text{maximum value of }\sum\limits_{k=1}^n\sin (kx)\right)$?

Define $f_n:[0,\pi]\to\mathbb{R}$ by $$ f_n(x) = \sum_{k=1}^{n} \sin(kx) = \frac{\sin(\frac{n}{2}x)\sin(\frac{n+1}{2}x)}{\sin(\frac{x}{2})}. $$ 1. Narrowing down the location of the maximum point. ...
Sangchul Lee's user avatar
20 votes
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A square contains many random points. From each point, a disc grows until it hits another disc. What proportion of the square is covered by the discs?

I did some literature search, and found that: OP's question exactly corresponds to what is called lilypond model introduced by Häggström and Meester (1996). Daley et al. (1999) independently studied ...
Sangchul Lee's user avatar
18 votes

What is the mean value of $|\sin x +\sin (\pi x)|$?

First, let's move from sum to product. $$\sin x + \sin \pi x = 2 \sin \frac{x(\pi +1)}{2} \cos \frac{x(\pi -1)}{2}$$ Therefore, $$\lim_{n \rightarrow \infty} \int\limits_{0}^n \frac{1}{n} |\sin x + \...
Denchik's user avatar
  • 326
18 votes
Accepted

How to solve $\displaystyle\lim_{n\to\infty}\int_0^3\underbrace{\sin(\frac{\pi}{3}\sin(\frac{\pi}{3}...\sin(\frac{\pi}{3} x)...))}_\text{n sines}dx$?

This answer gives an idea using fixed point iteration as $\sin(\frac\pi3x)$ is bounded and continuous, so nesting it will not create any singularities nor divergence and the limit should exist: $$\...
Тyma Gaidash's user avatar
17 votes

Evaluate $\lim_{n \to \infty} n \prod_{m = 1} ^ n \left(1 - \frac1m + \frac5{4m ^ 2}\right)$

We have the Weierstrass product for $\cos$: $$\cos(\pi z) = \prod_{m=1}^\infty\left(1-\frac{4z^2}{(2m-1)^2}\right)$$ valid for all $z\in \Bbb C$. Setting $z=i$ we get $$\cosh(\pi) = \prod_{m=1}^\infty\...
jjagmath's user avatar
  • 17.9k
17 votes

What is the sum of an infinite resistor ladder with geometric progression?

You can use the same trick as you did for the fixed factor case by observing that the resistance of the circuit $\hspace{10em}\text{Figure 1.}$ is exactly twice the resistance of the circuit $\...
lonza leggiera's user avatar
16 votes
Accepted

How to prove that $\lim_{x\rightarrow\infty}x-\sqrt{x(x+1)}=-\frac12$

\begin{align*} &\lim\limits_{x\rightarrow \infty} x-\sqrt{x(x+1)} \\ =& \lim\limits_{x\rightarrow \infty} \frac{x^2-x(x+1)}{x+\sqrt{x(x+1)}} \\ =& \lim\limits_{x\rightarrow \infty} - \frac{...
Sandipan Samanta's user avatar
16 votes
Accepted

Does the limit of these sums converge/diverge?

We have $$g(x)+if(x)=\sum_{n=0}^\infty {e^{ni}x^n\over n!}=e^{e^ix}=e^{x\cos 1+ix\sin 1}\\ =e^{x\cos 1}e^{ix\sin 1}$$ Then $$h(x)=|g(x)+if(x)|=e^{x\cos 1}$$
Ryszard Szwarc's user avatar
16 votes
Accepted

Why is the following limit not 1?

If $n\in\Bbb N$, then$$\frac{(n!)!}{(n!-n)!}=n!\times(n!-1)\times\cdots\times\bigl(n!-(n-1)\bigr)\geqslant n!,$$and therefore$$\lim_{n\to\infty}\frac{(n!)!}{(n!-n)!}=\infty.$$You cannot simply ignore ...
José Carlos Santos's user avatar
16 votes
Accepted

Possible error in Stanley's combinatorics volume 1

The exercise is false as printed, because $E(n)/n!\to \infty$ as $n\to\infty$. Stanley must have meant $E(n)/n^n\to 1/e$. This makes sense, because the author makes a comparison to $D(n)/n!\to 1/e$. ...
Mike Earnest's user avatar
  • 74.6k
16 votes

Is there any sequence of closed shapes whose limit tends to the unit circle while the limit of the perimeter goes to infinity?

Consider the polar curve: $$r=1+\frac1n\sin(n^2 \theta)$$ which for $n=2$ and $n=10$ looks like the following, corresponding to Ethan Bolker's suggestion. Clearly, as $n$ increases, the curve ...
Henry's user avatar
  • 156k
16 votes
Accepted

$x^{x^{x^{x^{x^{...}}}}} = 2$. Why is $-\sqrt{2}$ not a solution?

Short answer $x^{x^{x^{x^{x^{...}}}}}$ is not defined for $x < 0$. Explanation How is $x^{x^{x^{x^{x^{...}}}}}$ defined? It is the limit $\lim_{n\to\infty} a_n$ of the sequence \begin{align*}a_0 &...
azimut's user avatar
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15 votes
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When does this wrong method of solving integrals give the right answer?!

It comes down to the observation that the contribution to the PV integration from the immediate vicinity of the singularity was, in this problem, just zero. This follows from the fact that you can ...
Ian's user avatar
  • 101k
15 votes
Accepted

Why doesn't simplifying this limit work?

Properly: $\dfrac{\sin(x^2)}{\sin(x^2)+x^2\cos( x^2)}=\dfrac{1}{1+x^2(\cos(x^2)/\sin(x^2))}=\dfrac{1}{1+x^2\color{blue}{\cot(x^2)}}.$
Oscar Lanzi's user avatar
  • 38.7k
14 votes
Accepted

What is the mean value of $|\sin x +\sin (\pi x)|$?

This is not a direct answer $\require{AMScd}$ Funny enough, I cannot tackle the integral head on, but I do have a solution to the sum equivalent to that integral. Let's consider following question: ...
oO_ƲRF_Oo's user avatar
  • 1,235
14 votes
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Intuition is silent: Find the probability that the smallest circle enclosing $n$ random points on a disk lies completely on the disk, as $n\to\infty$.

First, let me state two lemmas that demand tedious computations. Let $B(x, r)$ denote the circle centered at $x$ with radius $r$. Lemma 1: Let $B(x,r)$ be a circle contained in $B(0, 1)$. Suppose we ...
abacaba's user avatar
  • 7,865
13 votes
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Why does limit: $\lim_{x\to\infty}\prod\limits^{\lfloor x\rfloor}_{n=1}\frac{\lceil x\rceil}{\lfloor x\rfloor}=e$?

Let $k=\lfloor x\rfloor$ $$\prod\limits^{\lfloor x\rfloor}_{n=1}\frac{\lceil x\rceil}{\lfloor x\rfloor}=\prod\limits^{k}_{n=1}\frac{k+1}{k}=\left( 1+\frac1k\right)^k$$ therefore, $$\lim_{x\to\infty\...
MathFail's user avatar
  • 21.1k
13 votes
Accepted

Finding the limit of $a_{n} = \frac{n!}{\left(\frac{2}{7} + 1\right)\left(\frac{2}{7} + 2\right)\ldots\left(\frac{2}{7} + n\right)}$

Note that $$ \frac{1}{a_n}=\frac{1}{n!}\prod_{k=1}^n\left(k+\frac{2}{7}\right)=\prod_{k=1}^n\left(1+\frac{2}{7k}\right). $$ Therefore we infer that $$ \log(1/a_n)=\sum_{k=1}^n\log\left(1+\frac{2}{7k}\...
imtrying46's user avatar
  • 2,587
12 votes
Accepted

Evaluate $ \lim_{n \to \infty} \sin^n\left(\frac{2\pi n}{3n+1}\right)$

For $n>2$, $$ \pi>\frac{2\pi}{3}>\frac{2\pi n}{3n+1}>\frac{2\pi n}{3n+n/2}=\frac{4\pi}{7}>\frac{\pi}{2}, $$ so for $n>2$, $$ \frac{\sqrt{3}}{2}=\sin\left(\frac{2\pi}{3}\right)<\...
Alexander Burstein's user avatar
12 votes
Accepted

Series which is a product of sequences which converges, but diverges when one of them is shifted by 1

Take $$a_{2n}=(-1)^n/n, a_{2n+1}=0, b_{2n}=1, b_{2n+1}=(-1)^n$$
Conrad's user avatar
  • 27k
12 votes
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What does this series converge to? $\sum_{n=0}^\infty \beta(4n+1)-\beta(4n+3)$

Recall that the Abel summation $\text{A-}\sum_{n=0}^{\infty} c_n$ is defined by $$ \text{A-}\sum_{n=0}^{\infty} c_n := \lim_{x \to 1^-} \sum_{n=0}^{\infty} c_n x^n. $$ We collect some properties of ...
Sangchul Lee's user avatar
12 votes
Accepted

Calculating pretty difficult limit that invloves Riemann sums

Extending the argument in this answer we can write $$\frac{1}{n}\sum_{k=1}^n f\left(\frac{k}{n}\right) =\int_0^1 f(x) \, dx+\frac{f(1)-f(0)}{2n}+\frac{f'(1)-f'(0)}{12n^2}+o(1/n^2)\tag{1}$$ Using $$f(x)...
Paramanand Singh's user avatar
  • 86.9k
12 votes
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Why does the limit $\lim _{a \rightarrow \infty} \frac{\int_0^a \sin ^4 x d x}{a}$ not exist?

The limit does exist. A slightly more general statement is as follows: Theorem. Suppose $f : \mathbb{R} \to \mathbb{C}$ is locally integrable (i.e., integrable on any finite subinterval of $\mathbb{...
Sangchul Lee's user avatar

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