An alternating series is a series whose terms alternate between positive and negative like a light switch being flipped on and off. Just like alternating sequences, the terms of such a series usually have a factor of (-1)n or (-1)n + 1 or something like that.
We will see that only some of these series can be tamed, while others run amok like a panther escaped from captivity in a city.
Sample Problem
Write the following sum using sigma notation.
-1 + 2 – 3 + 4 – 5 + 6 + ...
Answer.
Ignoring the fact that this is a series for a moment, look at the sequence of terms:
-1, 2, -3, 4, -5, 6,...
This is an alternating sequence. We know how to find the general term of an alternating sequence. This is a sequence whose nth term is (-1)n n, where the first term has n = 1. Now remember that we started with a series. We know the nth term and the starting index, so we can write the series in summation notation:
That's all there is to it.
For infinite series, we don't have to worry about "last terms" because there aren't any. We'll compute the first 3 or so terms, then write dots to show the pattern continues forever.
Sigma Notation
There's an important thing to understand about sigma notation: the letter we use for the index of summation doesn't matter. We can write
or
or
These all mean the same thing. If we wanted to use a little house for a summation index, we could. Sometimes we might use i for the index because we want to use n for the upper limit and write
Now that we've practiced translating series into sigma notation, let's practice translating series from sigma notation to expanded form. Going the other direction should be as easy as eating a slice of lemon meringue pie. Get your fork.
For short series we'll write all the terms. For longer series we'll write the first 2 or 3 terms and last 1 or 2 terms. We find the terms by plugging in the appropriate values of n to the general term, just like when we were finding specific terms of sequences.
Be Careful: When working with series, as with sequences, you should usually resist the urge to simplify. If you feel the urge, sit on your hands. If your friend feels the urge, throw a Pringle at them.
We know you've been told over and over to simplify your answers to everything. However, simplification can make it harder to spot the patterns. For example, consider the simplified series
and its un-simplified version
We think it's easier to see what's going on in the un-simplified version.