15.1 Infinite Series and Convergence

If we have an infinite sequence, we define it to be convergent if, for any positive criterion, q, however small, beyond some term, say the n(q)th, all of the terms are within q of some number, z which we call the limit of the sequence.

Where can you find infinite sequences?

Obviously we cannot write down each term of an infinite sequence in our lifetimes. Instead we can give a formula for the n-th term, and that does it.

For example, we can consider the sequence with n-th element is given by , or by , both of which have limits 0.

We can associate with any infinite series the sequence of its partial sums, the k-th partial sum being the sum of the first k of its terms. Then convergence of the series is defined to be the same as the convergence of that sequence of partial sums.

Here are some examples:

The "harmonic series"

The alternating harmonic series

The geometric series

If you were not asleep you would ask me: How can we tell whether or when a series is convergent?

Huh? How can we tell when a series is convergent?

By the way there are several kinds of convergence. There is ordinary convergence, already defined, and absolute convergence, which means convergence if we replace each term by a positive term of the same absolute value.

Then, if you have a series that is a function of a variable, like the geometric series given, there is uniform convergence, which means given a criterion q, there is an n(q) beyond which the partial sums to that point are within q of a limit value for all x in the given domain; the same n(q) independent of x.

If our first series above were convergent, then the second one would be absolutely convergent. The third series is uniformly absolutely convergent for |z| < a < 1, for any a, but not for |z| &ge 1.

Why do you care about absolute convergence?

When a series is absolutely convergent, you can rearrange its terms, differentiate it term by term if terms contain a variable, and perform other manipulations, which may not work for merely convergent series.

You can integrate it term by term if it is uniformly absolutely convergent between your limits of integration.

Fine. How can we tell when a series is convergent?