TITLE: Evaluate the sum $\lim_{t\to{1-0}}(1-t)(\frac{t}{1+t}+\frac{t^2}{1+t^2}+...+\frac{t^n}{1+t^n}+...)$
QUESTION [5 upvotes]: I have a question about this limit.
Calculate$$\lim_{t\to{1-0}}(1-t)(\frac{t}{1+t}+\frac{t^2}{1+t^2}+...+\frac{t^n}{1+t^n}+...)$$
Can anyone help?

REPLY [4 votes]: Let $h=-\log t$. Then
$$
\begin{eqnarray}
\lim_{t\nearrow 1}(1-t)\sum_{k=1}^\infty\frac{t^k}{1+t^k}
&=&
\lim_{t\nearrow 1}\sum_{k=1}^\infty(1-t)\frac1{1+t^{-k}}
\\
&=&
\lim_{h\searrow 0}\sum_{k=1}^\infty(1-\mathrm e^{-h})\frac1{1+\mathrm e^{kh}}
\\
&=&
\lim_{h\searrow 0}\sum_{k=1}^\infty(h+O(h^2))\frac1{1+\mathrm e^{kh}}
\\
&=&
\int_0^\infty\frac1{1+\mathrm e^x}\mathrm dx
\\
&=&
\log2\;.
\end{eqnarray}
$$