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7.2. Infinite Series

A related question is how to rigorously define the sum of infinitely many numbers, which is referred to as an infinite series. An example is the geometric series 1+x+x2+x3+…= 1/(1-x), which we used casually on page 29. The general concept of an infinite series goes back to ancient Greek mathematics. Various supposed paradoxes about infinite series, such as Zeno's paradox, were exhibited, influencing Euclid to sidestep the issue in his Elements, where in Book IX, Proposition 35 he provides only an expression (1-xn)/(1-x) for the nth partial sum of the geometric series. The case where n gets so big that xn becomes negligible is left to the reader's imagination, as in one of those scenes in a romance novel that ends with something like “...and she surrendered...” For those with modern training, the idea is that an infinite sum like 1+1+1+… would clearly give an infinite result, but this is only because the terms are all staying the same size. If the terms get smaller and smaller, and get smaller fast enough, then the result can be finite. For example, consider the geometric series in the case where x=1/2, for which we expect the result 1/(1-1/2) = 2. We have

\[1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + ... ,\]

which at the successive steps of addition equals 1, \(1\frac{1}{2}\), \(1\frac{3}{4}\), \(1\frac{7}{8}\), \(1\frac{15}{16}\), .... We're getting closer and closer to 2, cutting the distance in half at each step. Clearly we can get as close as we like to 2, if we're willing to add enough terms.

Note that we ended up wanting to talk about the partial sums of the series. This is the right way to get a rigorous definition of the convergence of series in general. In the case of the geometric series, for example, we can define a sequence of the partial sums 1, 1+x, 1+x+x2, ... We can then define convergence and limits of series in terms of convergence and limits of the partial sums.

It's instructive to see what happens to the geometric series with x=0.1. The geometric series becomes

1 + 0.1 + 0.01 + 0.001 + … .

The partial sums are 1, 1.1, 1.11, 1.111, ... We can see vividly here that adding another term will only affect the result in a certain decimal place, without affecting any of the earlier ones. For instance, if we needed a result that was valid to three digits past the decimal place, we could stop at 1.111, being assured that we had attained a good enough approximation. If we wanted an exact result, we could also observe that multiplying the result by 9 would give 9.999…, which is the same as 10, so the result must be 10/9, which is in agreement with 1/(1-1/10) = 10/9.

One thing to watch out for with infinite series is that the axioms of the real number system only talk about finite sums, so it's easy to get wrong results by attempting to apply them to infinite ones (see problem 2 on page 114).