Establishing identity: $\sum_{k = 1}^{\infty} \frac{1}{(2k)^{2}} + \sum_{k=1}^{\infty} \frac{1}{(2k+1)^{2}} = \sum_{k = 1}^{\infty} \frac{1}{k^{2}}$

by dc3rd   Last Updated August 14, 2019 00:20 AM

I need some help in trying to prove the following identity $$\sum_{k = 1}^{\infty} \frac{1}{(2k)^{2}} + \sum_{k=1}^{\infty} \frac{1}{(2k+1)^{2}} = \sum_{k = 1}^{\infty} \frac{1}{k^{2}}$$

I'm not sure where to start really. I thought it was going to be as simple as just adding the two terms together but nothing comes of that....

Answers 1

Just split the sum in RHS into the part where $k$ is even and the part where $k$ is odd.

Kavi Rama Murthy
Kavi Rama Murthy
August 14, 2019 00:19 AM

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