Computing $\lim_{\substack{t\to1\\t<1}}{(1-t)\sum_{n=0}^\infty{\frac{t^n}{1+t^n}}}$

Question

I want to find the following limit: $$ \lim_{\substack{t\to1\\t<1}}{(1-t)\sum_{n=0}^\infty{\frac{t^n}{1+t^n}}}.$$ I suspect it has to do with integrals and Riemann sums, but I don't see the exact way to proceed.

Thanks for help!