Mathematical Methods in the Physical Sciences · Problem 4.4

INFINITE SERIES, POWER SERIES — Mary Boas

Mathematics — Methods · convergent-divergent-series

Problem

Select some ϵ\epsilon's and find the corresponding NN's for the series ∑n=1∞1n!\displaystyle\sum_{n=1}^{\infty} \frac{1}{n!}. Hint: 1n!<12n\frac{1}{n!} < \frac{1}{2^n} when n>3n > 3 (see Problem 6.1); hence the remainder S−SnS - S_n is smaller than the remainder of a geometric series.

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