Doubt from sequence and series

please try to solve this i am not getting a proper answer


What is the answer?

no answer given my answer were not matching with the options

options were 1 2 1/4 and 1/2

do share your method please

The options are wrong,I'm pretty sure

What is your answer and please give your method

i am getting a range type of answer pi^2/12 +1/2 being the upper limit

lower limit i am getting 1/2 though

Let given expression equal S.

Then S= \displaystyle \sum_{n=1}^{\infty} \dfrac{1}{r^2} -\dfrac{2}{r^2+1}= \sum_{n=1}^{\infty}\dfrac{1}{r^2}- \sum_{n=1}^{\infty}\dfrac{1}{r^2+\frac{1}{2}}

= \dfrac{\pi^2}{6}- \displaystyle \sum_{n=1}^{\infty}\dfrac{1}{r^2+\frac{1}{2}}

We have in general {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}+t^{2}}}==-{\frac {1}{2t^{2}}}+{\frac {\pi }{2t}}\coth(\pi t).}

Here we substitute t=\dfrac{1}{\sqrt 2}