Lebl-Basic-Analysis-I · Problem 5.4.4

The Riemann Integral

Mathematics · logarithm-exponential

Problem

Use the geometric sum formula to show (for

t \neq -1

)

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) = \sum_{n=1}^\infty \frac{(-1)^{n+1} x^n}{n}

for all

x \in (-1, 1]

(note that

x = 1

is included). Finally, find the limit of the alternating harmonic series

\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots