# Linked Questions

19answers
56k views

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

How to prove $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$
18answers
7k views

### 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 ...
9answers
29k views

### 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 ...
2answers
801 views

### When are two proofs “the same”?

Often, we find different proofs for certain theorems that, on the surface, seem to be very different but actually use the same fundamental ideas. For example, the topological proof of the infinitude ...
8answers
8k views

### 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 ...
4answers
2k views

### 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?
3answers
416 views

3answers
127 views

### 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 ...
2answers
51 views

### Exponential problem in definite integral [closed]

$$\frac{1}{\sqrt{2\pi s^2}} \int_{-\infty}^{\infty} xe^{-(x-m)^2/(2s^2)} \, dx$$ I am stuck at this problem. Please give some hint as how to initiate?