The Alternating Series Test
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.
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.
Since the terms alternated sign the partial sums went up, then down, then up, like jumping back and forth over a line. Since the terms approached zero, the partial sums got closer and closer together, and closer and closer to whatever line they were jumping over.
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 an is an alternating series, and if
|an| > |an + 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.
Can we use the AST to conclude that the series
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 an 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 Sn is just an approximation of L. We'd like to know how good the approximation is—that is, how close are Sn and L? In symbols, we'd like to say something about the value.
|L – Sn|.
This distance between the approximation Sn 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 ai that sums to L. Then the error between Sn 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.