The Geometric Series

Abstract

If the harmonic series is the most celebrated of all divergent series, the same distinction for convergent series goes, without reservation, to the geometric series. We have already met this series in connection with the runner’s paradox. In a geometric sequence, or progression, we begin with an initial number a and obtain the subsequent terms by repeated multiplication by a constant number q: a, aq, aq², …, aqⁿ, …. The constant q is the common ratio, or quotient, of the progression. Sometimes our progression is terminated after a certain number of terms, in which case, of course, we omit the final dots. Such finite geometric progressions appear quite frequently in various situations. Perhaps the most well known is compound interest: If one deposits, say, $100 in a savings account that pays 5% annual interest, then at the end of each year the amount of money will increase by a factor of 1.05, yielding the sequence $100.00, 105.00, 110.25, 115.76, 121.55, and so on (all figures are rounded to the nearest cent).¹ On paper, at least, the growth is impressive; alas, inflation will soon dampen whatever excitement one might have derived from this growth!

With the exception of the geometric series, there does not exist in all of mathematics a single infinite series whose sum has been determined rigorously.

— Niels Henrik Abel (1802–1829)