Is divergence proven with ratio test if result is $1$?

Question

I have read about ratio test on the internet and it says if we got $1$ as result, we don't know whether the series converges or not.

So if I get $1$ as result in ratio test, I cannot say that this series diverges, right?

I will have to use something else then...? (direct comparison test & co.)

Answer 1

No. The case when the limit is $1$ is undertermined. For example using the ratio test on both $$\sum_n \frac 1 n \,\,\,\,\,\,\,\, \text{ and } \,\,\,\,\,\,\,\, \sum_n \frac 1 {n^2}$$ will both result in a limit of $1$ but one diverges and the other converges. When the ratio test fails, often times the comparison test or the integral test work. The integral test is the one I normally go to next.

Answer 2

In case the ratio test fails because the limit is equal to $1$, you have a refinement with Raabe-Duhamel's rule, which uses asymptotic analysis:

Suppose $\dfrac{u_{n+1}}{u_n}=1-\dfrac\alpha n+o\Bigl(\dfrac1n\Bigr)$. Then:

• if $\alpha<1$, the series diverges;
• if $\alpha>1$, the series converges;
• if $\alpha=1$, we cannot conclude.