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.

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

Definition.

By ageometric serieswe mean any series of the formfor some real constants

aandq.

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*q* = 1*q* = −1*q* = 1/2*q* different from 1):

For *q* = 1 we have
*s*_{N} = *N* + 1,*s*_{N} is a sum of *N* + 1*N* we get the following statement.

Fact.

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

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.

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 *k*^{q}? The
reason is that we want to have *p* > 0

Indeed, when we look at a *p*-series (in the above form) and ask
what happens to its terms when *p* < 0,*k*^{p}}*p* = 0*p* > 0,*k*^{p}*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
*k*^{p}*p* = 0*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.

Fact(p-test).

Ifp> 1, then the series converges.

Ifp≤ 1, then .

The proof of this extremely useful *p*-test*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
*k*^{2}

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),**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 *H*_{n}. 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.

See also this note.

**Remark:**
This is a good opportunity to emphasize one key difference. When we have
a bunch of numbers *a*_{k}, 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 *k*

**Remark** concerning subseries: Note that the series in the first
example (the one with *k*^{2})*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.

By an alternating series we mean any series whose terms have alternating
signs, that is, the signs in this series go
^{k}*b*_{k}*b*_{k} positive. (See also
Alternating
sequence in Sequences - Theory - Introduction - Important examples).

Note that sometimes we have a
natural expression of the series that has
^{k+1}*b*_{k}^{k}*b*_{k}

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.