Let and be any two fixed real numbers with The series
is called the geometric series with first term and ratio
Notice that we have chosen to start the summation index at That’s fine. The first term is the term, which is The second term is the term, which is And so on. We could have also written the series That’s exactly the same series — the first term is the second term is and so on . Regardless of how we write the geometric series, is the first term and is the ratio between successive terms.
Geometric series have the extremely useful property that there is a very simple formula for their partial sums. Denote the partial sum by
The secret to evaluating this sum is to see what happens when we multiply it by
Notice that this is almost the same as The only differences are that the first term, is missing and one additional term, has been tacked on the end. So
Hence taking the difference of these expressions cancels almost all the terms:
Provided we can divide both side by to isolate
On the other hand, if then
So in summary:
Now that we have this expression we can determine whether or not the series converges. If then tends to zero as so that converges to as and
On the other hand if diverges. To understand this divergence, consider the following 4 cases:
In each case the sequence of partial sums does not converge and so the series does not converge.
Here are some sketches of the graphs of and for and
In these sketches we see that
when and also when with odd, we have On the other hand, when with even, we have
When gets closer and closer to as increases.
When just alternates between when is odd, and when is even.