A series is an infinite sum of the terms of sequence.

In the previous sections, we’ve seen several examples of sequence. If we have a sequence , and represent it as an ordered list below:

we can ask two important questions about it.

  • Do the numbers in the list approach a finite value?
  • Can I sum all of the numbers in the list and obtain a finite result?

The first question is really whether the limit exists and we studied several ways to determine this previously. As it turns out, the second question is closely related to the first one. We motivate this through an example.

Infinite series

The above scenario can be modeled using sequences. We have whose -th term is given by the explicit formula , and we represent the sequence by the ordered list below.

and we can interpret the series as the attempt to add up all of the terms in the above sequence. In order to determine if this can be done, we found a new sequence that was created from the original sequence. It is represented by the list below.

Note that the limit of this new sequence is exactly the sum of all of the terms in the old sequence! Let’s formalize the ideas in the last example with a definition.

The above definition really states that the symbols and are exactly the same! It makes the content of the previous example more precise. This is quite important since we have techniques to determine whether limits of sequences exist. We are now able to recast the new question “Can I sum all of the terms in a sequence?” into the old question “Does the associated sequence of partial sums have a limit?”. To attack the latter, we can utilize all of our previous techniques for studying the limit of the sequence of partial sums.

Using our new terminology, what can we say about the series from the previous example? The series convergesdiverges , and .

Now, let’s see an example.

Properties of sums

We finish this section by giving some properties of series.

Notice, of course, that we’re working with convergent series in this theorem. Adding divergent series is trickier, but there is something we can say about the attempt to add a convergent series and a divergent series.

To understand why this theorem holds, note that if would converge, by the last theorem, we would know that the difference

converges since both series above are convergent. Furthermore, the previous theorem also guarantees that

but this is precisely , which diverges by assumption. This is a contradiction, so must diverge.

Essentially, this theorem ensures that it does not matter where we start summing the terms of a sequence. Since we have infinitely many terms to try to add together, the sum of the first finitely many will not affect if this addition is possible.

We now finish the section with an example that ties many ideas together.