# Important examples

Here we will show two basic examples of series - the geometric series and the p-series. They are quite important, since many tests of convergence use these two types as benchmarks. At the end we look at alternating series.

## Geometric series

We obtain geometric series by summing up a geometric sequence (see Sequences - Theory - Introduction - Important examples).

Definition.
By a geometric series we mean any series of the form

for some real constants a and q.

This is the most general definition, but many people prefer to work with series of the form . In fact, knowing such series is enough, since any series as in our definition can be easily transformed into a series of this simpler form. Indeed, we can obviously take the constant a and factor it out of the series, so it does not influence its behavior and at the end it can be easily factored into whatever result we get without it. The second problem to handle is to transform the indexing so that it starts at 0. This can be done by factoring the lowest power out of the series and then reindexing using substitution.

We can see better why it works if we write the series in the long way, then it is actually obvious.

As we can see, this factoring out trick depends heavily on the fact that terms of this series are all powers with the same base, in other words, it is not going to work with other series than geometric (or those very similar to them). Anyway, since the more general form can be handily changed into the easier one and it is bother to write, in the sequel we will always use the more friendly, simpler form of a geometric series.

Geometric series is very popular, since we know all about it. Note that in fact all examples in the first section (see Introduction) were geometric series. Indeed, the choice q = 0 gives the series , the choice q = 1 gives the series , and the choice q = −1 gives the series . Finally, the choice q = 1/2 gives the really interesting series . So what can we say about geometric series? First, using mathematical induction we easily prove that for partial sums we have the following formula (for q different from 1):

For q = 1 we have sN = N + 1, the plus one part is there because we start indexing at 0, so sN is a sum of N + 1 numbers 1. Now passing to infinity with N we get the following statement.

Fact.
Consider a geometric series  .  This series converges for |q| < 1 and diverges for |q| ≥ 1. Moreover,

We have already shown that it is enough to know formulas for series whose index starts at 0, but some people prefer not to play with algebra and remember more general formulas instead.

There are six basic types of geometric series, they correspond to the types described in Important examples in Sequences - Theory - Limit.

## Series with powers ( p-series)

In this part we will consider series of the form , where p is a parameter. A natural question that immediately arises is this: Why do we write this series in this form, with terms as fractions; why don't we write those terms in the simpler form kq? The reason is that we want to have p > 0 so that we do not have to mess around with signs.

Indeed, when we look at a p-series (in the above form) and ask what happens to its terms when p < 0, then the answer is obvious. The sequence {1/kp} goes infinity and therefore the series diverges by the necessary condition. Similarly, for p = 0 we obtain a divergent series whose terms are all 1. Thus the only remaining case where there is a chance for convergence of the series is when p > 0, since then the terms 1/kp tend to zero. Thus with the choice of terms as we did, we are working almost exclusively with positive p. Note however that for negative integers p we are sometimes very interested in partial sums, see the next section.

What is happening when we consider positive p as advertised? Note that when we increase p, then the numbers 1/kp get smaller; this means that we are summing up smaller numbers and the series has a better chance of converging. Imagine that we start with p = 0 where the series diverges and start increasing p, thus increasing our hopes for convergence. Do we get lucky or are all p-series divergent? It turns out that as we increase p, then there is a border value where divergence changes into convergence.

Factp-test).
If p > 1, then the series   converges.
If p ≤ 1, then .

The proof of this extremely useful p-test follows easily from Integral test, see Theory - Testing Convergence. While it is important to remember this test, some series are useful enough that it is a good idea to remember them by heart. Since they are so popular, we will outline elementary proofs of their convergence/divergence without using the p-test.

Example: Consider the series .
The p-test says that this series is convergent p = 2 > 1).

In the next section we will talk about how difficult it is to sum up a series. This series is no different, it takes quite a bit of work to show that in fact

Since all terms of this series are positive, it automatically converges absolutely and therefore we also have convergence for all modifications of signs. In particular, if we change this series into an alternating series, we get convergence and also another interesting result.

Bonus: Elementary proof that the series of 1/k2 converges.

Now it seems that the series should converge and the sum should not exceed 2. This is indeed true, a rigorous proof can be obtained by a slight refinement of the above estimate (see telescopic series in the next section and comparison test in Theory - Testing convergence).

Example: Consider the series .
The p-test says that this series is divergent p = 1), in fact it is the borderline case. It is fairly important and it is called the harmonic series since its terms give relative wavelengths of harmonic tones on a string. In particular, every its term is the harmonic mean of the two neighboring terms. We will remember that

The partial sums of the harmonic series are called harmonic numbers and denoted Hn. Their precise values are not known, despite quite a bit of research went into learning more about them. The harmonic series is rather interesting, since once we turn it into an alternating series, it becomes convergent, see the Alternating series test (in Theory - Testing convergence). We even know what the sum is (see the next section),

This alternating series converges, but when we enclose its terms in absolute values, we obtain the divergent harmonic series. Thus this is the prime example of a conditionally convergent series.

Bonus: Elementary proof that the harmonic series diverges. This proof is said to be one of the high points of medieval mathematics.

Remark: This is a good opportunity to emphasize one key difference. When we have a bunch of numbers ak, we can form two distinct objects from them, a sequence and a series. These two need not have the same properties. For instance, the numbers 1/k as a sequence converge (to 0), but when we turn them into a series, it is divergent (the harmonic series above). Therefore it is important to specify exactly what we mean. This is in particular important in case when we do work both with series and with sequences (which is very often the case when testing convergence of a series), quite often one can see the statement "it converges" and it is not clear whether "it" refers to a sequence or a series. If you always say "the sequence diverges/converges," "the series diverges/converges," you should be safe.

Remark concerning subseries: Note that the series in the first example (the one with 1/k2) is a subseries of the harmonic series. Indeed, we can take the harmonic series and then sum up only its terms with coefficients k from the set A of squares, obtaining the first series. This shows that we can start with a divergent series and get a convergent series out of it by choosing just some terms. Actually, this sounds quite feasible, if a series diverges because we attempted to add too large numbers, we definitely improve our chances by dropping some of them.

A bit more surprising is the fact that it can also work the other way around. For instance, we know that the alternating harmonic series is convergent. However, if we take every second term in it, we get the divergent series

The proof that it diverges is easy using limit comparison.

## Alternating series

By an alternating series we mean any series whose terms have alternating signs, that is, the signs in this series go ...+ − + − + − +... Formally we say that alternating series are series that can be expressed as  ∑ (−1)kbk  with all bk positive. (See also Alternating sequence in Sequences - Theory - Introduction - Important examples).

Note that sometimes we have a natural expression of the series that has (−1)k+1bk instead of (−1)kbk (see examples above). However, this is no problem, such a series is also alternating, in particular because we can use reindexing to make it conform to the "proper" definition. We show one such reindexing in an example on Alternating series test in Theory - Testing convergence.

Alternating series appear quite often and they are (relatively) simple to handle, witness the Alternating series test that we already mentioned or the last Fact in Approximating series in Theory - Introduction to Series.