54k views

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

How to prove $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$
6k 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 ...
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 ...
7k 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 ...
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?
794 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 ...
For a freshman calculus project, I used Euler's approach to find $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$, and noted from Wikipedia's explanation that the infinite product representation of $\frac{... 3answers 1k views ### Evaluatig:$\int_{0}^{\infty}{e^{ax^2}\cos(bx)dx}$Evaluatig: $$\int_{0}^{\infty}{e^{ax^2}\cos(bx)dx}$$ Where$a, b\in \mathbb R^+$What i have done: Because$\cos(bx)=\Re(e^{ibx})\$, we can note that: $$I=\Re{\int_{0}^{\infty}{e^{ax^2+ibx}}}dx=\Re{\... 3answers 206 views ### Does there exist a group that is both a free product and a direct product of nontrivial groups? Do there exist such nontrivial groups A, B, C and D, such that A \times B \cong C \ast D? I failed to construct any examples, so I decided to try to prove they do not exist by contradiction.... 3answers 102 views ### What is the largest possible variance of a random variable on [0; 1]? What is the largest possible variance of a random variable on [0; 1]? It is evident that it does not exceed 1, but I doubt, that 1 is actually possible. The largest variance, for which I found ... 3answers 127 views ### Is there a way to sum up the series give below?? Is there a way to sum up the series with nth term$$ x_n=1/n^2$$I am high school student and I tried my level best to find a method to sum it up but failed.If there is a way to find this sum can this ... 3answers 206 views ### Rearranging infinite product I know that$$\frac{\sin x}x=\prod_{n=1}^\infty \left(1-\frac{x^2}{n^2\pi^2}\right).$$Why exactly can I take the product and factor x^2?$$\prod_{n=1}^\infty \left(1-\frac{x^2}{n^2\pi^2}\right)=1-...
$$\frac{1}{\sqrt{2\pi s^2}} \int_{-\infty}^{\infty} xe^{-(x-m)^2/(2s^2)} \, dx$$ I am stuck at this problem. Please give some hint as how to initiate?