Closed form of $\sum_{n=1}^{\infty} \frac{1}{2^n(1+n^2)}$ How would you recommend me to tackle the series 
$$\sum_{n=1}^{\infty} \frac{1}{2^n(1+n^2)}$$?
Can we possibly express it in terms of known constants? What do you think about it?

You may recall the Lerch transcendent function
$$
\Psi(z,s,a)=\sum_{k=0}^\infty\frac{z^k}{(a+k)^s}
$$
and use
$$
\frac{1}{n^2+1}=\frac{i}{2}\left(\frac{1}{n+i}-\frac{1}{n-i}\right)
$$
to get
$$
\sum_{n=1}^{\infty} \frac{1}{2^n(1+n^2)}=-1-\Im \: \Psi\left(\frac12,1,i\right)
$$
which gives your series in terms of a known special function. 
Maybe someday we will be able to say something deeper...
