Convergence of Series - At A Glance

Now that we know how to bake an infinite brownie, we want to know what it tastes like. We're 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.

Sample Problem

Find the 4th partial sum of the series

Answer.

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.

Sample Problem

The 1st partial sum of the series

If you think a series seems like the evil twin of a sequence, you're almost right. A series is the evil cousin of the sequence. They're 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 a_n of the series:
a₁, a₂, a₃,...
The sequence whose terms are the partial sums S_n of the series:
S₁, S₂, S₃,...