More tests for series with non-negative terms

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.

Raabe's test

One of the most popular convergence tests for series with positive terms is the Ratio test. It is inconclusive if we have λ = 1. There are more refined tests that may help, they look closer at how exactly do the fractions a_k+1/a_k approach 1. As an example we show one little-known test that does not even have a name.

Theorem.
Consider a series  ∑ a_k  such that a_k > 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 a_k+1/a_k tend to 1. It can be stated in many ways, we start with the simplest to use.

Theorem (Raabe's test - limit version).
Consider a series  ∑ a_k  such that a_k > 0 for all k. Assume that the limit

converges.
• If ϱ > 1 then the series  ∑ a_k  converges.
• If ϱ < 1 then the series  ∑ a_k  diverges.

Again, the case ϱ = 1 yields no information. For an example see this problem in Solved Problems - Testing convergence.

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  ∑ a_k  such that a_k > 0 for all k.
• If there is some A > 1 and some integer N so that for all k > N we have

then the series  ∑ a_k  converges.
• If there is some real number M and some integer N so that for all k > N we have

then the series  ∑ a_k  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  ∑ a_k  such that a_k > 0 for all k. Assume that there exists a real number A and numbers v_k such that for every k one has

and the series  ∑ v_k  is absolutely convergent.
Then the series  ∑ a_k  converges if and only if 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 they use a_k/a_k+1. Then the connection with the Ratio test is not as close, but it fits in with a more general picture. Indeed, Raabe's test is just a special version of Kummer's test where we traditionally work with the other 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  ∑ a_k  such that a_k > 0 for all k.
• It converges if and only if there is some A > 0, positive numbers p_k and an integer N such that for all k > N we have

• It diverges if and only if there are some positive numbers p_k 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  ∑ a_k  such that a_k > 0 for all k. Assume that for some positive numbers p_k the limit

converges.
• If ϱ > 0 then the series  ∑ a_k  converges.
• If ϱ < 0 and  , then the series  ∑ a_k  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 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 for all k, then we get the Ratio test. If we decide to take p_k = k for all 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 compare to 1 + 1/k? Some tests address this question.

Theorem (Gauss's test).
Consider a series  ∑ a_k  such that a_k > 0 for all k. Assume that there is a real number A, a number r > 1 and a bounded sequence {B_k} such that for all k we have

Then the series  ∑ a_k  converges if and only if 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 such that B_k = D_k⋅k ^r are bounded. If we find such 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 instead of a_k+1/a_k as above, we would need precisely the limit for 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  ∑ a_k  such that a_k > 0 for all k. Let numbers ϱ_k satisfy (for all k)

• If  liminf(ϱ_k) > 1 then the series  ∑ a_k  converges.
• If  limsup(ϱ_k) < 1 then the series  ∑ a_k  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  ∑ a_k  such that {a_k} 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  ∑ a_k. Let f be a non-increasing positive function such that a_k = f (k). Assume that the limit

converges.
• If r < 1 then the series  ∑ a_k  converges.
• If r > 1 then the series  ∑ a_k  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 g(x) > x is the condition to have, and then 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
Back to Theory - Testing convergence