Question

True or False? \[\sum_{k=odd}^{}\frac{ 1}{ k^2 }=\left(\begin{matrix}\frac{ \pi }{ 2 }\end{matrix}\right)^2\]
I know the answer, but I am wondering what different approaches people might take to quickly check and then write a proof.

Answer 1

\[\zeta(2) = 1+\frac{1}{2^2}+\frac{1}{3^2}+ \frac{1}{4^2}+\cdots = \dfrac{\pi^2}{6}\]
\[\dfrac{\zeta(2)}{2^2} = \frac{1}{2^2}+\frac{1}{4^2}+ \frac{1}{6^2}+\cdots = \dfrac{\pi^2}{24}\]
\[\zeta(2)-\dfrac{\zeta(2)}{2^2} =1+ \frac{1}{3^2}+\frac{1}{5^2}+\cdots = \dfrac{\pi^2}{6}-\dfrac{\pi^2}{24}\]

Answer 2

What is this zeta function represent?

Answer 3

\(\zeta(n) = \sum\limits_{k=1}^{\infty} \dfrac{1}{k^n}\)

Answer 4

https://en.wikipedia.org/wiki/Riemann_zeta_function

Answer 5

Cool thanks

Answer 6

I already found a use for it, you can use this for QM expectation value of the energy

Answer 7

Now the problem boild down to proving \[\zeta(2) = \dfrac{\pi^2}{6}\]

Answer 8

consider this\[\zeta(-1) = 1 + 2 + 3 + 4 + \cdots = -\frac{1}{12}\]

Answer 9

interesting..

Answer 10

this guy in 12th grade figured out his own proof lol https://brilliant.org/discussions/thread/proof-that-zeta2dfracpi26/

Answer 11

My favorite is the proof by euler

Answer 12

Notice that the function \(f(x) = \sin x \) is \(0\) iff \(x = n \pi\), \(n\in \mathbb Z\)

Answer 13

There's one using fourier series to https://en.wikipedia.org/wiki/Basel_problem

Answer 14

therefore \((1-\dfrac{x}{n\pi})\) is a factor of \(\sin x\) :

\[\sin x = x\prod\limits_{n\in\mathbb{Z}_{\ne 0}} (1-\dfrac{x}{n\pi})\]

Answer 15

which is same as

\[\begin{align}\sin x &= x\prod\limits_{m\in\mathbb{N}} (1-\dfrac{x}{m\pi}) (1+\dfrac{x}{m\pi})\\~\\ &=x\prod\limits_{m\in\mathbb{N}} (1-\dfrac{x^2}{m^2\pi^2})\end{align}\]

Answer 16

Ask a question, and a world of knowledge pours in. Thank you for the input and great ideas.