Ratio Test Intuition

Question

The limit of the ratio as n approaches infinity, when 1, results in an inconclusive ratio test.

The way I am trying to understand the ratio test (without a proof but instead by intuition) is that when the next term n+1 is less than term n, the ratio is less than 1 and the series is convergent, because as n becomes very large, the terms become smaller and smaller and smaller.

But by this logic, if the limit of the ratio is 1, the series should be divergent, because the term n and term n+1 will be equal as n approaches infinity. It's like adding 1 + 1 + 1 + 1 + 1 ... and will diverge.

I know that the ratio test is inconclusive when 1 as there are examples of convergent and divergent series having the ratio 1, but I would like to clarify this intuition issue.

Answer 1

I think you are using the wrong intuition. Terms getting smaller and smaller does not imply convergence. It doesn't even imply that the terms go to zero, and going to zero is not enough for convergence either, as the harmonic series demonstrates. The intuition you want is that the terms in your series are eventually always less than those of a convergent geometric series. When the limit of the ratio is $1$, you don't have such a convergent geometric series to compare to.

Intuition can be changed by by studying key examples, and I think that's what has to be done in this situation since untrained intuition is notoriously unreliable when it comes to infinite series. The key examples you need are

1. geometric series with ratio $<1$ (convergent),
2. harmonic series: $\sum_{n=1}^\infty\frac{1}{n}$ (divergent and ratio goes to 1),
3. $\sum_{n=1}^\infty\frac{1}{n^p}$ with $p>1$ (convergent and ratio goes to 1).

If you want intuition, think of it this way: terms can go to $0$, but each stay close enough to the next successive term that the ratio goes to $1$. So you can have ratio going to $1$, and have it still be like adding up a bunch of terms close to $0$. But being close to $0$ isn't enough for convergence since you're adding infinitely many terms. What's needed for convergence is that terms go to $0$ fast enough.

One more thing, adding a bunch of terms whose ratio is approaching $1$ isn't like adding a constant repeatedly, $$ \epsilon+\epsilon+\epsilon+\epsilon+\ldots. $$ Even though nearby terms may be nearly equal and may all be close to some constant $\epsilon$, terms that come much later in the series can be smaller than $\epsilon$ by an arbitrarily small ratio. (That's what it means for terms to be approaching $0$.)