Section Summary

In this section we have developed three tests for convergence of series with infinitely many terms. If the series is a power series with variable `x`, each of these tests leads to an interval of values of `x` for which the series converges. If `x` has already been assigned a value (i.e., we are testing a series of constants), then each test may or may not yield an answer about convergence.

In the order considered, our tests are

• the Alternating Series Test (AST),
• Comparison to a Geometric Series (CGS), and
• the Ratio Test (RT).

In a supplementary activity associated with this chapter you may have an opportunity to explore another test,

• the Integral Test (IT).

The order of presentation was appropriate for development of new ideas from familiar ones, but it is not the order in which one would usually use these tests. Confronted with a power series whose radius of convergence is unknown, your best starting point is the Ratio Test, which is just a reorganization and simplification of CGS. It identifies an interval in which the power series definitely converges. If that interval is the entire number line, there is nothing else to do. If the interval has endpoints, the test also tells us that the series diverges outside those endpoints. That leaves only the endpoints to be tested. The Divergence Test is the next thing to think about — if the terms of an endpoint series don't go to zero, it can't converge. If an endpoint series has alternating signs and terms that go to zero in a disciplined way, AST tells you that it converges. If the terms do not alternate in sign, you can check to see if the series is a multiple of a series whose convergence or divergence is known, such as the harmonic series. You also can attempt to apply IT.