While there are quite a few tests for series with non-negative terms, most of that is lost once we allow some sign changes in the series. There are very few tests that would work for general series. We start with those, then we look at the Alternating series test, discuss absolute convergence and conclude this section with Abel's test and Dirichlet's test.

The simplest general test is the

Necessary condition for convergence.

If a sequence {a_{k}} does not tend to zero, then the series∑ diverges.a_{k}

(See Basic properties in Theory - Introduction.) However, this is only an implication and unfortunately it helps only very rarely. More useful (although definitely not universally applicable) are versions of the popular Root and Ratio tests for general series.

Theorem(Root test for general series).

Consider a series∑ Assume that the limita_{k}.converges.

• Ifthen the series converges. ϱ< 1

• Ifthen the series diverges. ϱ> 1

Theorem(Ratio test for general series).

Consider a series∑ Assume that the limita_{k}.converges.

• Ifλ < 1 then the series converges.

• Ifλ > 1 then the series diverges.

(See the Note at the end of the section Root and Ratio tests in Theory - Testing convergence.) As usual, when the two constants end up being 1, then we cannot say anything about the given series.

Often we have a case when the signs in the series change ...+ − + − +... (that is, the series is alternating) and the terms decrease in absolute value. In that case the situation is the best possible, since then we have a simple test of convergence that gives complete information.

Theorem(Alternating series test, AST).

Consider a series that can be expressed as∑ (−1) for some positive^{k}b_{k}b_{k}that form a non-increasing sequence.

This series converges if and only if, the sequence{ tends to 0.b_{k}}

Note that one direction in this theorem is trivial. If the numbers
*b*_{k} do not go to zero, then also the terms of the
series
*a*_{k} = (−1)^{k}*b*_{k}*a*_{k} do go to
zero, then in general we cannot conclude anything, but in the alternating
situation as above it is already enough to show convergence.
That is the important thing about the above test.

Note that the above test also applies to alternating series whose signs
go in the opposite way, formally
^{k+1}*b*_{k}*b*_{k} (that is, we subtract every even term
instead of every odd term in the series). Indeed, given such a series, we
can factor one minus sign out of the whole series, then we get a new series
whose signs go according to the pattern in the above test and whose
convergence is equivalent to convergence of the given series.

**Example:** Investigate convergence of the series

We see that every term of the given series can be expressed as
^{k+1}*b*_{k}*b*_{k} = 1/*k* > 0.^{k}*b*_{k},*k*}

Note that we could have changed the given series into a "proper alternating series" also by reindexing like this:

Then we would use the AST with
*b*_{n} = 1/(*n* − 1).

Some examples can be found in Solved Problems - Testing convergence, namely this problem and this problem.

In more advanced courses students learn some less popular tests that can help also with other combinations of signs than the alternating situation. At the end of this section we cover Dirichlet's test that can be though of as a generalization of AST for more complicated patterns of signs.

Of course, in a situation when terms in a series change signs one would have
to be really lucky to get an alternating series, since there are many ways
in which the signs may change. Then we cannot use AST, even if the changes
follow a regular pattern (for instance two pluses always followed by one
minus or something like this). Note that in the

If we find that the series
*a*_{k}|*a*_{k}

However, it may also happen that the series with absolute values diverges and then we do not know anything about convergence of the given series. There are no general guidelines concerning such a situation, every series is an individual problem and all depends on our experience, we try to find some trick to decide its convergence.

Sometimes one can use one of the following two tests.

The following test allows us to take a series that we know to converge and modify it without losing its convergence.

Theorem(Abel's test).

Consider a convergent series∑ Assume thata_{k}.{ is a monotone and convergent sequence. Then also the seriesb_{k}}∑ converges.a_{k}b_{k}

In fact, Abel's test can be deduced from the following, more general criterion.

Theorem(Dirichlet's test).

Consider a series∑ such that all its partial sums are bounded; that is, there exists somea_{k}Mso that for allNone hasAssume that {

b_{k}} is a non-increasing sequence of positive numbers that tends to 0. Then the series ∑a_{k}b_{k}converges.

For examples see this problem and this problem in Solved problems - Testing convergence.

This test can be used in a wide range of situations, but one particular
situation is worth exploring closer. Note that this test also implies the
Alternating series test above. Indeed, consider the situation as in the
assumptions of AST. We have some positive numbers
*b*_{k} that are non-increasing and go to zero. Now we
form an alternating series from them, which we can interpret as follows: We
create a series with terms
*a*_{k}*b*_{k},*a*_{k} = (−1)^{k}.*a*_{k} satisfy
assumptions of the Dirichlet test, the same is true for numbers
*b*_{k} and we get convergence.

We have just explained why Dirichlet's test generalizes the AST, but it is
interesting to return to this situation and explore it a bit more. We claim
that when we start with positive numbers *b*_{k}
that are non-increasing and go to zero, then the Dirichlet test allows us to
modify them by signs also in other ways than just alternating. Indeed,
assume that terms *a*_{k} are numbers −1 and 1 that are
chosen in such a way that a certain pattern is repeated over and over (a
periodic sequence so to speak). If in that basic pattern there is equal
number of minuses and pluses, then partial sums can again attain only
finitely many distinct values, so they are bounded and the Dirichlet test gives
convergence. This is definitely a significant improvement over the rather
special Alternating series test. We conclude this section by stating this
generalization of AST properly.

Fact.

Consider numbersAssume that there is a_{k}= ±1.such that T> 0for all natural numbers a_{kT+i}=a_{i}kand integersisatisfying 0≤ Leti<T.

a_{1}+a_{2}+ ... +a_{T}= 0.Assume that

{ is a non-increasing sequence of positive numbers that tends to 0. Then the seriesb_{k}}∑ converges.a_{k}b_{k}