We discuss convergence results for geometric series and telescoping series.

*series*and have to determine whether it converges or diverges. To answer this question, we define a new

*sequence*where for all . We saw previously that

- the series
**converges**if and only if exists. - the series
**diverges**if and only if does not exist.

The most straightforward way to determine whether exists is to have an explicit formula for the -th term . Note that this is not an easy task; for example, can you find a formula for for the series ? It’s not too hard to write out the first several terms in the sequence , but try to find an explicit formula that describes the next term in the list.

### A recursive formula for

As it turns out, there is always a recursive formula for . For the sake of example, suppose that we want to consider . Let’s write out the formula for .

We can make an observation by considering in a similar way.

Now returning to our expression for , we can make an observation.

We thus have the formula

If we apply this to the series , where , the result read . This does not help
us analyze whether actually exists. Sometimes, however, we can find an
*explicit* formula for , and we study two special types of series for which this is
possible.

### Geometric series

Recall that a *geometric sequence* is a sequence for which the ratio of successive terms
is constant. If is such a sequence, then there are constants and for which
.

This generates the ordered list

and we have a result for the limits of these types of sequences, which we now recall.

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.

**sequence of partial sums**of .

- (a)
- The series
**converges**if and only if exists. Furthermore, if , we say the series converges to . - (b)
- The series
**diverges**if and only if or otherwise does not exist.

The above definition really is assuring us that the symbols and are exactly the same! However, this definition makes the content of the previous example more precise. The major idea here is that we have techniques that we can use to determine whether limits exist and can even find what those limits are sometimes. Since we are now able to recast the new question “Can I sum all of the terms in a sequence?” into the old question “Does a sequence have a limit?”, we can now utilize all of our previous techniques to analyze the sequence of partial sums.

### Two special types of series

The definitions above give us a way to determine whether a given series converges. In fact, to determine whether converges, we can do the following.

- Consider the associated sequence of partial sums.
- Try to find an explicit formula for the term . If you can find such a formula,
analyze .
- If the limit exists, converges, and if we can determine that , then .
- If does not exist, then diverges.

- If an explicit formula for cannot be found, further analysis is needed. We’ll expound on this in later sections.

We can now think of adding together the terms of a geometric sequence.

Before exploring when such a series converges, note that sometimes, some preliminary algebra is necessary to recognize a series as geometric.

Using the laws of exponents shows us:

Indeed, and .

We can now try to determine when adding together the terms in such a
series is possible; that is, we can explore for which values of and the *series*
converges.

The issue with finding a formula for arises from the fact that we cannot perform the above for an unspecified value . However, to go from one term in the sum to the next, we multiply by , so let’s multiply both sides of the above equation by .

Now, we can subtract away the middle terms. We can now solve for .

Since is **always** a finite sum, and , there is no issue with manipulating it the way we
did.

From our work above, we see that the -th partial sum of the geometric series is We
now have an *explicit* formula so we can determine for which values of the limit
exists. First, note that the limit in question, is the limit of a *geometric* sequence. In
fact,

In fact, if , the is not bounded above. If , the is bounded, but the terms oscillate between and . If , the terms both oscillate in sign and become arbitrarily large in magnitude.

The above formula covers every case except when , but notice that so if , and , so diverges.

When , note , so in this case,

By noting that , we can combine this observation with the above argument and write the result in a theorem.

There is a useful trick that allows us to find the sum of a convergent geometric series when the lower index does not start at .

This is now a geometric series whose lower index is , so we can use the formula to find its value. Noting that and gives:

We can easily generalize this example and doing so allows us to write down a more comprehensive theorem about geometric series.

This matches the earlier result!

The lower index in a series does not affect whether the series converges or diverges, but if the series converges, it can affect the value to which the series converges.

The formula listed above very explicitly shows exactly how the lower index affects the value to which a convergent geometric series converges.

Now, try some questions to check your understanding of the above material.

### Telescoping series

A second type of series for which we can find an explicit formula for are “telescoping series”. Rather than try to give a formal definition, we think of telescoping series are infinite sums for which the required addition required to find a formula for can be done so many of the intermediate terms naturally cancel. An example will make this point more clear.

*all*terms in the sequence can be done formally by using an idea called

*mathematical induction*. We leave it to the curious reader to explore this idea further if desired.

We’ve just seen an example of a **telescoping series**. Informally, a telescoping series
is one in which the partial sums reduce to just a finite sum of terms. In
the last example, the partial sum only was the sum of two nonzero terms:

### Summary

Now that we have seen two special types of series for which we can find an explicit formula for the -th term in the sequence of partial sums, it helps to summarize the logic that we employed.

- Consider the associated sequence of partial sums.
- Try to find an explicit formula for the term . If you can find such a formula,
analyze .
- If the limit exists, converges, and if we can determine that , then .
- If does not exist, then diverges.