Mathematical Methods in the Physical Sciences · Problem 7.10

INFINITE SERIES, POWER SERIES — Mary Boas

Mathematics — Methods · alternating-series

Problem

The following alternating series are divergent (but you are not asked to prove this). Show that

a_n \to 0

. Why doesn't the alternating series test prove (incorrectly) that these series converge? (a)

2 - \frac{1}{2} + \frac{2}{3} - \frac{1}{4} + \frac{2}{5} - \frac{1}{6} + \frac{2}{7} - \frac{1}{8} \cdots

(b)

\frac{1}{\sqrt{2}} - \frac{1}{2} + \frac{1}{\sqrt{3}} - \frac{1}{3} + \frac{1}{\sqrt{4}} - \frac{1}{4} + \frac{1}{\sqrt{5}} - \frac{1}{5} \cdots