Introduction
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The theory of infinite series is a third branch of calculus, in addition to differential and integral calculus. Infinite series yield a new perspective on functions and on many interesting numbers. Two examples are the infinite series for the exponential function \[ e^x = 1 + x +\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots \] and the Gregory–Leibniz series (see Exercise 53 in Section 2) \[ \frac{\pi}4 = 1 - \frac13+\frac15-\frac17+\frac19-\cdots \] The first shows that \(e^x\) can be expressed as an “infinite polynomial,” and the second reveals that \(\pi\) is related to the reciprocals of the odd integers in an unexpected way. To make sense of infinite series, we need to define precisely what it means to add up infinitely many terms. Limits play a key role here, just as they do in differential and integral calculus.
