Question

Using 1/z^2sin(pi z) prove that sum of (-1^n)/n^2 =pi^2/12

Answer 1

Step-by-step explanation:

I'm doing a homework problem, and so far I've proved

∑n=−∞∞1(z+n)k=(−2πi)k(k−1)!∑m=1∞mk−1e2πimz

for k an integer ≥2 and Im(z)>0. The next part of the problem asks me to put k=2 and show

∑n=−∞∞1(z+n)2=π2sin2(πz)

for Im(z)>0, but after nearly an hour of bashing I still see it-- I've tried product expansions of sine, using Euler's formula (which equates to showing e2πir+e−2πir−216=−4π2∑∞m=1me2πimz). Could anybody point out a way to get the above equality from the one I derived? Apologies in advance if I'm just ditzy and it's really obvious.

Also, the next question asks if the above formula is true if z is any complex number that is not an integer, but I'm not really sure I understand what it's asking. Why wouldn't it be true?