Linked Questions

177
votes
19answers
56k views

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

How to prove $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$
94
votes
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 ...
61
votes
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 ...
38
votes
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 ...
23
votes
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 ...
12
votes
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?
9
votes
3answers
416 views

Question about Euler's approach to find $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$

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{...
5
votes
3answers
219 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....
3
votes
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{\...
1
vote
3answers
109 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 ...
1
vote
3answers
215 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-...
0
votes
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 ...
0
votes
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?