Convergence of Series
Now that we know how to bake an infinite brownie, we want to know what it tastes like. We are going to need to find a cow big enough to give us a glass of milk to match.
In terms of series, this means adding up infinitely many numbers? We would expect the sum of an infinite list of numbers to be infinite. Some of the time, we'd be wrong.
Before we can understand how we can fit an infinite sum into a finite box, we have to define something called a partial sum. A partial sum is what we get if we add up some of the terms of a series. This is like breaking off a part of our infinite brownie and sharing it with friends.
More specifically, the nth partial sum of a series is the sum of the first n terms of the series.
Find the 4th partial sum of the series
To find the 4th partial sum we add the first 4 terms:
The 4th partial sum is .
The 1st partial sum of a series is its first term.
The 1st partial sum of the series
If you think a series seems like the evil twin of a sequence, you are almost right. A series is the evil cousin of the sequence. They are related through a sequence of partial sums
S1, S2, S3, S4,... where the nth term is the nth partial sum of the series.
Be Careful: Remember that sequences and series are different things.
In reality, a series is the evil triplet of two different sequences. Any series
has two different sequences associated with it:
- The sequence whose terms are the same as the terms an of the series:
a1, a2, a3,...
- The sequence whose terms are the partial sums Sn of the series:
S1, S2, S3,...
These two sequences are different.