# History Of Infinite Series

What is a good source for the history of infinite series?

Moreover, why do we learn them?

Are they really useful on their own, or are just tools / stepping stone for studying series of functions later?

• – Mark Fantini Sep 20 '14 at 9:52
• Thank you Mark Fantini – Vagabond Sep 25 '14 at 9:32
• Does the history of infinite series converge? – rbp Oct 13 '14 at 13:25
• The references I cited in my comment to the math StackExchange question References for mathematical theory of summability of divergent series might provide some useful references to the history you're looking for. – Dave L Renfro Oct 13 '14 at 17:58
• When you use your calculator to calculate a function like a sine or an exponential, it's calculating it using a series. – Ben Crowell Oct 17 '14 at 19:04

## 1 Answer

Of course infinite series are useful. Consider three early examples:

• Euclid (~300 BC) found the geometric series $\sum 1/r^n$ useful, and contemporary applications would have included Archimedes's quadrature of the parabola and for Zeno's paradoxes.

• Oresme (~1350) found the harmonic series $\sum 1/n$ useful, as an example of a divergent sum.

• Mercator (1668) found $\sum \pm\, x^n/n$ as a useful series for calculating logarithms $\log(1+x)$.

There are plenty of historical sources on any of those three topics, though not necessarily connected to each other. Also I wouldn't focus on the history too much, since contemporary applications to compound interest may be a better hook.

In any case, people used infinite series well before there were Taylor series or series of functions in much generality. The same series, with updated interpretations, still provide good reasons to learn the topic today.