In my research, I encountered a "two sided" series similar to the one
below:

sum_{n = -infinity}^{infinity} a_n , a_n=[r^n/(1+r^(2*n))] , r<1

{i.e., Series going from -infinity to +infinity which has nth term as

[r^n/(1+r^(2*n))] }

Is there a closed form expression for the sum of this series? Is it
possible to find the sum of the this series as we have for a geometric
series.