Linked Questions

375
votes
15answers
46k views

Why does $1+2+3+\cdots = -\frac{1}{12}$?

$\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$. Why should analytically continuing to $\zeta(-1)$ give the right answer?
284
votes
101answers
26k views

Surprising identities / equations

What are some surprising equations/identities that you have seen, which you would not have expected? This could be complex numbers, trigonometric identities, combinatorial results, algebraic results, ...
67
votes
4answers
4k views

A strange integral: $\int_{-\infty}^{+\infty} {dx \over 1 + \left(x + \tan x\right)^2} = \pi.$

While browsing on Integral and Series, I found a strange integral posted by @Sangchul Lee. His post doesn't have a response for more than a month, so I decide to post it here. I hope he doesn't mind ...
48
votes
3answers
15k views

Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$

Using $\text{n}^{\text{th}}$ root of unity $$\large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
44
votes
7answers
2k views

Bad Fraction Reduction That Actually Works

$$\frac{16}{64}=\frac{1\rlap{/}6}{\rlap{/}64}=\frac{1}{4}$$ This is certainly not a correct technique for reducing fractions to lowest terms, but it happens to work in this case, and I believe there ...
21
votes
2answers
710 views

On the “funny” identity $\tfrac{1}{\sin(2\pi/7)} + \tfrac{1}{\sin(3\pi/7)} = \tfrac{1}{\sin(\pi/7)}$

This equality in the title is one answer in the MSE post Funny Identities. At first, I thought it had to do with $7$ being a Mersenne prime, but a little experimentation with Mathematica's integer ...
9
votes
1answer
407 views

How is it possible that $\infty!=\sqrt{2\pi}$?

I read from here that: $$\infty!=\sqrt{2\pi}$$ How is this possible ? $$\infty!=1\times2\times3\times4\times5\times\ldots$$ But \begin{align} 1&=1\\ 1\times2&=2\\ 1\times2\times3&=6\\ &...
3
votes
0answers
301 views

Proof that $ 1^3+2^3+\cdots +n^3 = (1+2+\cdots+n)^2$ without using induction [duplicate]

Possible Duplicate: Intuitive explanation for the identity $\sum\limits_{k=1}^n {k^3} = \left(\sum\limits_{k=1}^n k\right)^2$ How to prove this without using mathematical induction? $$1^3+2^3+\...
2
votes
0answers
170 views

How can the following “funny identity” be generalised?

When asked for a "funny identity", Andrey Rekalo answered the following: $$\left(\sum\limits_{k=1}^n k\right)^2=\sum\limits_{k=1}^nk^3 .$$ Not only do I think it's funny, I also think it's very ...
1
vote
2answers
167 views

Help with Trigonometry homework - prove an identity

I need to prove the following identity: $\sin^2 2\alpha-\sin^2 \alpha = \sin 3\alpha \sin \alpha$ What I have tried, is to work on each side of the identity. I have started with the left side: \...
1
vote
0answers
56 views

How to prove $\int_0^1\frac{\mathrm{d}x}{x^x}=\sum_{k=1}^\infty \frac1{k^k}$? [duplicate]

I've found this wonderful identity here. On the first spot it seems for me true, but I don't have any idea, how could it be proven. In simpler form: "how the hell...?" :-)