Mathematical Methods in the Physical Sciences · Problem 6.2

INFINITE SERIES, POWER SERIES — Mary Boas

Mathematics — Methods · comparison-test

Problem

Prove that the harmonic series

\displaystyle\sum_{n=1}^{\infty} 1/n

is divergent by comparing it with the series

1 + \tfrac{1}{2} + (\tfrac{1}{4}+\tfrac{1}{4}) + (\tfrac{1}{8}+\tfrac{1}{8}+\tfrac{1}{8}+\tfrac{1}{8}) + (8 \text{ terms each equal to } \tfrac{1}{16}) + \cdots,

which is

1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \cdots