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### Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \frac{\sqrt \pi}{2}$

How to prove $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$
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### What is the most unusual proof you know that $\sqrt{2}$ is irrational?

What is the most unusual proof you know that $\sqrt{2}$ is irrational? Here is my favorite: Theorem: $\sqrt{2}$ is irrational. Proof: $3^2-2\cdot 2^2 = 1$. (That's it) That is a ...
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### Proof that a Combination is an integer

From its definition a combination $\binom{n}{k}$, is the number of distinct subsets of size $k$ from a set of $n$ elements. This is clearly an integer, however I was curious as to why the expression ...
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### Need to prove the sequence $a_n=1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}$ converges

I need to prove that the sequence $a_n=1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}$ converges. I do not have to find the limit. I have tried to prove it by proving that the sequence is monotone ...
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### Is $(\mathbb{Q},+)$ the direct product of two non-trivial subgroups?

Is this statement true or false? I am really not having any idea how to prove or a counterexample, please help. Is $(\mathbb{Q},+)$ a direct product of two non-trivial subgroups?
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### Is there a way to sum up the series give below??

Is there a way to sum up the series with nth term $$x_n=1/n^2$$ I am high school student and I tried my level best to find a method to sum it up but failed.If there is a way to find this sum can this ...
$$\frac{1}{\sqrt{2\pi s^2}} \int_{-\infty}^{\infty} xe^{-(x-m)^2/(2s^2)} \, dx$$ I am stuck at this problem. Please give some hint as how to initiate?