The source for this problem is this 3b1b video. I understand this:
$$\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...=\frac{\pi^2}{6}$$
Now he alters the series to include only primes and powers of primes (eg. 4 and 8 are included because they are powers of 2, which is prime) while scaling down the powers of primes by a factor of the exponent, as in:
$$\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2(2)}+\frac{1}{5^2}+\frac{1}{7^2}+\frac{1}{8^2(3)}+\frac{1}{9^2(2)}+...$$
This happens to equate to: $$ \ln\left(\frac{\pi^2}{6}\right)$$
I tried to search the proof of this for a while, but could not find anything. I apologize if this happens to be a very trivial question for this site, but would be delighted to see an elementary explanation because I am just an interested layman in this matter.
from Hot Weekly Questions - Mathematics Stack Exchange
Bipasha
Post a Comment