Infinite Series

As we saw in the previous chapter, many calculus problems have a solution that can be expressed as an infinite series. It is therefore useful to be able to recognize important individual series and to understand their general properties and capabilities. This is the aim of the present chapter. Starting with the infinite geometric series, already known to Euclid, we discuss the handful of examples known before the invention of calculus. These include the

harmonic series

$$1 + 1/2 + 1/3 + 1/4 + \cdot\cdot\cdot$$

, studied by Oresme around 1350, and the stunning series for the inverse tangent, sine, and cosine, discovered by Indian mathematicians in the 15th century. The invention of calculus in the 17th century released a flood of new series, mostly of the form

$$a_0 + a_1x + a_2x^2 + \cdot\cdot\cdot$$

(called

power series

), but also some variations, such as fractional power series. The 18th century brought new applications. De Moivre (1730) used power series to find a formula for the

n

th term of the

Fibonacci sequence

0, 1, 1, 2, 3, 5, 8,.... Euler (1748a) introduced a generalization of the harmonic series,

$$1 + 1/2^s + 1/3^s + 1/4^s + \cdot\cdot\cdot$$

, and showed that, for

s

> 1, it equals the

infinite product

$$(1 - 1/{2^s})^{-1}(1 - 1/3^s)^{-1}(1 - 1/5^s)^{-1} \cdot\cdot\cdot (1 - 1/p^s)^{-1} \cdot\cdot\cdot$$

over all the

prime

numbers

p

. This discovery of Euler’s opened a new path to the secrets of the primes, exploration of which continues to this day.