Here we gathered some tests that are undoubtedly useful, but they are less convenient than the tests covered in other sections and as such they are skipped in most calculus courses. After Raabe's test we look at Kummer's test and some of its consequences, at the end we introduce Cauchy's condensation test and Ermakoff's test.

One of the most popular convergence tests for series with positive terms is
the Ratio test. It is
inconclusive if we have
*a*_{k+1}/*a*_{k}

Theorem.

Consider a series∑ such thata_{k}for all a_{k}> 0k.

• This series converges if• This series diverges if there is an integer

Nsuch that for allwe have k>N

Now we pass to the most popular test used in cases when
*a*_{k+1}/*a*_{k}

Theorem(Raabe's test - limit version).

Consider a series∑ such thata_{k}for all a_{k}> 0k. Assume that the limitconverges.

• Ifthen the series ϱ> 1∑ converges.a_{k}

• Ifthen the series ϱ< 1∑ diverges.a_{k}

Again, the case *ϱ* = 1

Just like with the Root test and the Ratio test, the assumption on the existence of the limit above may be too much. Some versions handle this by using limsup, others give up on limit entirely and check on individual fractions. One possible version goes like this.

Theorem(Raabe's test).

Consider a series∑ such thata_{k}for all a_{k}> 0k.

• If there is someand some integer A> 1Nso that for allwe have k>Nthen the series

∑ converges.a_{k}

• If there is some real numberMand some integerNso that for allwe have k>Nthen the series

∑ diverges.a_{k}

As we hinted, there are quite a few modifications of this test around. This one is among the most general.

Theorem(Raabe's test).

Consider a series∑ such thata_{k}for all a_{k}> 0k. Assume that there exists a real numberAand numbersv_{k}such that for everykone hasand the series

∑ is absolutely convergent.v_{k}

Then the series∑ converges if and only ifa_{k}A>1.

We see that here we have an equivalence, which is pretty strong for a
convergence test. Note that all the above tests can be also stated in such a
way that instead of the ratio
*a*_{k+1}/*a*_{k}*a*_{k}/*a*_{k+1}.

What comes now is probably the most powerful convergence test, in particular because its statements are not implications but equivalences.

Theorem(Kummer's test).

Consider a series∑ such thata_{k}for all a_{k}> 0k.

• It converges if and only if there is somepositive numbers A> 0,p_{k}and an integerNsuch that for allwe have k>N• It diverges if and only if there are some positive numbers

p_{k}such that and an integerNsuch that for allwe have k>N

While the equivalences are indeed impressive, this generality makes the test rather unwieldy in practice. A somewhat simpler (but less powerful) version uses limit.

Theorem(Kummer's test - limit version).

Consider a series∑ such thata_{k}for all a_{k}> 0k. Assume that for some positive numbersp_{k}the limitconverges.

• Ifthen the series ϱ> 0∑ converges.a_{k}

• Ifand , then the series ϱ< 0∑ diverges.a_{k}

Obviously there are tough series where we are glad to have the powerful
general version of Kummer's test, but for less fiendish series it is an
overkill. The hard part is coming up with numbers
*p*_{k} that will make this test work, since their
choice can be very tricky. For simpler series we prefer to use some less
powerful but more user-friendly corollaries. In particular, if we decide to take
*p*_{k} = 1*k*,
then we get the Ratio test. If we decide to take
*p*_{k} = *k**k*, then we get Raabe's test.

There are other ways to weaken the Kummer's test. A deeper analysis shows
that the critical question is this: How does
*a*_{k}/*a*_{k+1}*k*?

Theorem(Gauss's test).

Consider a series∑ such thata_{k}for all a_{k}> 0k. Assume that there is a real numberA, a numberand a bounded sequence r> 1{ such that for allB_{k}}kwe haveThen the series

∑ converges if and only ifa_{k}A> 1.

Here we also have some strange numbers *B*_{k}, so what
is the advantage over the Kummer test? While in Kummer's test we have to
somehow guess those *p*_{k}, in Gauss's test there is
a reasonable procedure to get to the appropriate constants. The best
*A* can be obtained by a limit and then we check on what remains, namely
we set

Then we try to find *r* > 1*B*_{k} = *D*_{k}⋅*k* ^{r}*r*, we are ready to use the Gauss test.
For an example see
this problem in Solved Problems -
Testing convergence.

By the way, if we wanted a version of Raabe's test that uses
*a*_{k}/*a*_{k+1}*a*_{k+1}/*a*_{k}*A* that we just used.

The following test is sometimes also called deMorgan's and Bertrand's test.

Theorem(Bertrand's test).

Consider a series∑ such thata_{k}for all a_{k}> 0k. Let numberssatisfy (for all ϱ_{k}k)• If

liminf( then the seriesϱ_{k}) > 1∑ converges.a_{k}

• Iflimsup( then the seriesϱ_{k}) < 1∑ diverges.a_{k}

This test is even more straightforward, we simply calculate those rho's and then check on them.

It often so happens that the terms of a given series decrease to zero. The convergence or divergence of this series is then decided by how fast do the terms go. One way to take advantage of this situation is to realize that this speed of convergence need not be tested everywhere, it is enough to check just someplace ("jump" in the series).

Theorem(Cauchy's condensation test).

Consider a series∑ such thata_{k}{ is a non-increasing sequence of positive numbers.a_{k}}

This series converges if and only if the series converges.

The following test also looks closer at how fast the terms go to zero.

Theorem(Ermakoff's test).

Consider a series∑ Leta_{k}.fbe a non-increasing positive function such thatAssume that the limit a_{k}=f(k).converges.

• Ifthen the series r< 1∑ converges.a_{k}

• Ifthen the series r> 1∑ diverges.a_{k}

Also this test has a version that sidesteps the problem of convergence of the limit there , namely for convergence of the series it is enough that limsup is less than 1, while for divergence of the series it is enough that liminf is greater than 1.

A more general version exist, instead of working with the exponential in the
limit above one can in fact choose any increasing and differentiable function
*g* that grows fast enough, namely
*g*(*x*) > *x**r* is obtained as a limit of
*g*′(*x*)*f* (*g*(*x*))/*f* (*x*).

For an example of the above two tests being used see this problem in Solved problems - Testing convergence.

Convergence of general series

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convergence