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.
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'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:
These two sequences are different.
We can use the idea of partial sums to (finally) describe what it means to add infinitely many numbers together. We can taste our infinite brownie, too. In symbols, we want to know what
means, and just how much chocolaty goodness we're going to get.
Think about what happens when you put laundry detergent into a washing machine. As you add more suds, there's enough to wash the laundry and all goes well. You can also add so much that a detergent monster oozes from the washer, soaping your dry socks and cat Binx.
The same happens with series. As we add numbers, if the sum gets closer and closer to some particular value L, this means the partial sums
S1, S2, S3, ...
get closer and closer to L. In this case, it makes sense to say that when we add those infinitely many numbers together, we get L.
If the sum doesn't get closer and closer to any particular number as we keep adding and adding, there's no reasonable way to say what the sum of those infinitely many numbers is. The series is an untamable detergent beast.
Since we promised food, we can put this in terms of a brownie, or a brownie and limits. Specifically, we're talking about the limit of the sequence of partial sums.
Start with a series and look at the associated sequence of partial sums
S1, S2, S3, ....
As with any sequence, this sequence may converge or diverge.
If the sequence of partial sums converges to a finite number L, then we say the series converges to L. In symbols, the series converges to L if there's a finite number L with
In this case we say that L is the sum of the series.
If the sequence of partial sums doesn't converge, then the series doesn't converge either, and we say the series diverges.
In terms of a brownie, if our brownie continues to grow so large that it won't fit inside any box, then the brownie is infinite. But if there is a box it will fit inside of, the brownie is finite in size.
Be Careful: From here on, we're using the word converge to mean three or four different things.
The word converge means different things depending on whether we're talking about sequences, series, functions, or improper integrals.
Finding the sum of a series can be difficult, because not every series has a nice formula for the partial sum, Sn. Fortunately, we often only care if a series converges or diverges. That's much easier than finding the exact sum. We'll see later that we have a number of tools in a handy, leather tool belt to help us figure these things out.
The finite series
3 + 4
can be rewritten as the infinite series
3 + 4 + 0 + 0 + 0 + 0 + ...
The partial sums of this series are
S1 = 3
S2 = 7
S3 = 7
and Sn = 7 for all larger n. The sequence of partial sums is
3, 7, 7, 7, 7, 7,...
Since the sequence of partial sums converges to 7, it makes sense to say that the series
3 + 4
converges to 7.
It turns out that the sequence of terms and the sequence of partial sums of a series are not distant second cousins. They're both born from the same mother series, so they're siblings. Sometimes they look just alike, similar to identical twins. Sometimes they look as different as peas and carrots. They go together, nonetheless.
Look at any series
We can make a sequence out of the terms of the series:
a1, a2, a3, ...
We can also make a sequence out of the partial sums of the series:
S1, S2, S3, ....
These two sequences are different. They won't necessarily converge to the same value. They might not both converge. However, they are related in one key way.
If the sequence of partial sums converges, then the sequence of terms must converge to zero.
We know that saying, "the sequence of partial sums converges," is the same as saying, "the series converges." Here's a another mind-blowing fact.
If a series converges, then its terms must converge to zero.
Mathematician and magician are two very different professions. Because we don't like waving our hands, turning around three times and spitting to prove things, we're going to show both of these are true using math.