More about Infinite Series

Infinite geometric sequences and series arise not only in pure mathematics but also in geometry, physics, and engineering, and at least one contemporary artist, Maurits C. Escher, has based on them many of his works. We shall examine some of these in the following chapters. Meanwhile, let us take a brief look at some other series, several of which mark important milestones in the history of mathematics. We have seen that the harmonic series 1 + 1/2 + 1/3 + 1/4 + … diverges. But the corresponding series with the squares of the natural numbers has baffled mathematicians for many years; among them were several of the Bernoulli brothers, who all failed to find its sum, although it had been known for some time that the series converges.1 It was the great Swiss mathematician Leonhard Euler (1707–1783) who finally solved the mystery.