# Tag Info

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

Take $$a_{2n}=(-1)^n/n, a_{2n+1}=0, b_{2n}=1, b_{2n+1}=(-1)^n$$
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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 ...
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### 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)...
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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{...