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
Theorem.
Consider a series∑ ak such thatak > 0 for all k.
• This series converges if• This series diverges if there is an integer N such that for all
k > N we have
Now we pass to the most popular test used in cases when
Theorem (Raabe's test - limit version).
Consider a series∑ ak such thatak > 0 for all k. Assume that the limitconverges.
• Ifϱ > 1 then the series∑ ak converges.
• Ifϱ < 1 then the series∑ ak diverges.
Again, the case
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∑ ak such thatak > 0 for all k.
• If there is someA > 1 and some integer N so that for allk > N we havethen the series
∑ ak converges.
• If there is some real number M and some integer N so that for allk > N we havethen the series
∑ ak diverges.
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∑ ak such thatak > 0 for all k. Assume that there exists a real number A and numbers vk such that for every k one hasand the series
∑ vk is absolutely convergent.
Then the series∑ ak converges if and only ifA >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
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∑ ak such thatak > 0 for all k.
• It converges if and only if there is someA > 0, positive numbers pk and an integer N such that for allk > N we have• It diverges if and only if there are some positive numbers pk such that and an integer N such that for all
k > N we have
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∑ ak such thatak > 0 for all k. Assume that for some positive numbers pk the limitconverges.
• Ifϱ > 0 then the series∑ ak converges.
• Ifϱ < 0 and , then the series∑ ak diverges.
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
pk 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
There are other ways to weaken the Kummer's test. A deeper analysis shows
that the critical question is this: How does
Theorem (Gauss's test).
Consider a series∑ ak such thatak > 0 for all k. Assume that there is a real number A, a numberr > 1 and a bounded sequence{Bk} such that for all k we haveThen the series
∑ ak converges if and only ifA > 1.
Here we also have some strange numbers Bk, so what is the advantage over the Kummer test? While in Kummer's test we have to somehow guess those pk, 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
By the way, if we wanted a version of Raabe's test that uses
The following test is sometimes also called deMorgan's and Bertrand's test.
Theorem (Bertrand's test).
Consider a series∑ ak such thatak > 0 for all k. Let numbersϱk satisfy (for all k)• If
liminf(ϱk) > 1 then the series∑ ak converges.
• Iflimsup(ϱk) < 1 then the series∑ ak diverges.
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∑ ak such that{ak} is a non-increasing sequence of positive numbers.
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∑ ak. Let f be a non-increasing positive function such thatak = f (k). Assume that the limitconverges.
• Ifr < 1 then the series∑ ak converges.
• Ifr > 1 then the series∑ ak diverges.
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
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