Mar 21 - Introduction to Series

We talked about series today. A series means "an infinite sum, something of the form a + b + c + ..." with the dot, dot, dot implying that the pattern continues forever.

Determining the behavior of a series will be the fundamental problem of the next two weeks. We'll learn a variety of tests to explain that the sum is findable (the series converges) or that the sum is not findable (the series diverges). Today's two main ideas are the geometric series and the behavior of the nth term.

We did geometric series last year. Identify the constant ratio for the series as r and then if | r |< 1, the series converges. Easy.

The nth term should be mostly intuitive. If I say what is the sum of "2 + 2 + 2 + ..." you would recognize that the sum will continue to increase and thus goes to infinity (the series diverges). The nth term test says that if the limit of the terms in the series aren't approaching 0, then the series diverges.

Note that we don't get to say that if the terms are approaching 0, then the series converges. Turns out that's not necessarily true. Stay tuned.

Notes from today are below.