Mathematical Methods in the Physical Sciences · Problem 6.6

INFINITE SERIES, POWER SERIES — Mary Boas

Mathematics — Methods · comparison-test

Problem

There are 9 one-digit numbers (1 to 9), 90 two-digit numbers (10 to 99). How many three-digit, four-digit, etc., numbers are there? The first 9 terms of the harmonic series 1+12+13+⋯+191 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{9} are all greater than 110\frac{1}{10}; similarly consider the next 90 terms, and so on. Thus prove the divergence of the harmonic series by comparison with the series [110+110+⋯ (9 terms each=110)]+[90 terms each=1100]+⋯=910+90100+⋯=910+910+⋯\left[\tfrac{1}{10}+\tfrac{1}{10}+\cdots\,(9\text{ terms each}=\tfrac{1}{10})\right]+\left[90\text{ terms each}=\tfrac{1}{100}\right]+\cdots = \tfrac{9}{10}+\tfrac{90}{100}+\cdots = \tfrac{9}{10}+\tfrac{9}{10}+\cdots

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