Basic Analysis I — Real Analysis · Problem 5.4.4

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · logarithm-series

Problem

Use the geometric sum formula to show (for t≠−1t \neq -1) 1−t+t2−⋯+(−1)ntn=11+t−(−1)n+1tn+11+t.1 - t + t^2 - \cdots + (-1)^n t^n = \frac{1}{1+t} - \frac{(-1)^{n+1}t^{n+1}}{1+t}. Using this fact show ln⁡(1+x)=∑n=1∞(−1)n+1xnn\ln(1+x) = \sum_{n=1}^\infty \frac{(-1)^{n+1} x^n}{n} for all x∈(−1,1]x \in (-1,1] (note that x=1x=1 is included). Finally, find the limit of the alternating harmonic series ∑n=1∞(−1)n+1n=1−12+13−14+⋯\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots

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