IMC 2001 · Day 1 · P3

$$\begin{align*} \lim_{t \to 1-0} (1-t) \sum_{n=1}^{\infty} \frac{t^n}{1 + t^n} &= \lim_{t \to 1-0} \frac{1-t}{-\ln t} \cdot (-\ln t) \sum_{n=1}^{\infty} \frac{t^n}{1 + t^n} = \\ &= \lim_{t \to 1-0} (-\ln t) \sum_{n=1}^{\infty} \frac{1}{1 + e^{-n \ln t}} = \lim_{h \to +0} h \sum_{n=1}^{\infty} \frac{1}{1 + e^{nh}} = \int_0^{\infty} \frac{dx}{1 + e^x} = \ln 2. \end{align*}$$