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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?

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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.

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