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### 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....
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### 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 ...
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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 ...
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 $\frac{... 8answers 8k views ### Need to prove the sequence$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},+)$a direct product of two non-trivial subgroups? 9answers 29k views ### Proof that a Combination is an integer From its definition a combination$\binom{n}{k}$, is the number of distinct subsets of size$k$from a set of$n$elements. This is clearly an integer, however I was curious as to why the expression ... 19answers 56k views ### Proving$\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \frac{\sqrt \pi}{2}\$
How to prove $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$