The Alternating Series Test - At A Glance

The first tool in our arsenal of convergence tests is for alternating series, which is a series whose terms alternate in sign. This is for a Pandora's box full of American and Swiss grilled cheese sandwiches. The alternating series test can tell us if it's safe to open that box.

Sample Problem

The series

1 – 2 + 3 – 4 + 5 – 6 + ...

is an alternating series because the signs of the terms switch back and forth between positive and negative.

We can find a formula for the terms of an alternating series the same way we can find a formula for the terms of an alternating sequence.

Line jumping is the idea behind our first convergence test, the alternating series test. Since the terms of an alternating series change sign, the partial sums for any alternating series will jump back and forth over some line. If the terms are getting smaller and approaching zero, the partial sums will get closer to the line and so the series will converge.

Alternating Series Test (AST): If Σ a_n is an alternating series, and if

|a_n| > |a_{n + 1}|

for all n (that is, the terms have strictly decreasing magnitude), and if

then the series converges.

The catch: We can't use the AST to conclude a series diverges. We can only use the AST to conclude that a series converges. If the AST doesn't tell us that the series converges, we need to use another test, which might include the divergence test.

Sample Problem

Can we use the AST to conclude that the series

converges?

Answer.

We have an alternating series. For all n, we have

so the first condition is met. We also have

so the second condition is met. Since all conditions are met, the AST says that the series converges.

Suppose we know, now, that we have an alternating series Σ a_n that converges, and it converges to value L. Here, L stands for limburger, which is the type of stinky cheese in our Pandora's box. We know we can open it without fear of a sandwich explosion, but it will smell terrible.

A partial sum S_n is just an approximation of L. We'd like to know how good the approximation is—that is, how close are S_n and L? In symbols, we'd like to say something about the value.

|L – S_n|.

This distance between the approximation S_n and the real sum of the series L is called the error. With alternating series, it's awesomely easy to find an upper bound for the error.

In general, suppose we have a convergent alternating limburger series Σ a_i that sums to L. Then the error between S_n and the actual sum L can't be more than the absolute value of the (n + 1)st term of the series.

We use the absolute value of the (n + 1)st term because the only thing the sign is good for is saying which way the jump goes.

To find the (n + 1)st term of the series we need to know the starting index of summation, or we won't know which term is the (n + 1)st.