A rule of thumb for when to use the Ratio Test

The Ratio Test is a powerful test for both sequences and series of non-zero real numbers. It comes in various forms, but here is one commonly used version.

Theorem (Ratio Test)
Let $(x_n)_{n\in\N}$ be a sequence of non-zero real numbers. Suppose that

\[\frac{|x_{n+1}|}{|x_n|}\to L \text{ as } n\to\infty\,,\]

where $L$ is either a non-negative real number or $+\infty$.

(a) If $L>1$, then $|x_n| \to +\infty$ as $n \to + \infty$, the sequence $(x_n)$ is divergent, and the series $\displaystyle \sum_{n=1}^\infty x_n$ diverges.

(b) If $L \in [0,1)$, then the series $\displaystyle \sum_{n=1}^\infty x_n$ is absolutely convergent (and hence convergent), and the sequence $(x_n)$ converges to $0$.

(c) If $L=1$, then the Ratio Test is inconclusive, and you need to use a different test. (The Ratio Test tells you nothing in this case about the convergence or otherwise of either the sequence or the series.)

Notes

Here (b) does not tell you anything about the value of the sum of the series $\displaystyle \sum_{n=1}^\infty x_n$. But we do know that if a series converges, then the sequence of terms must tend to $0$. (The converse to this useful fact is false, as is shown, for example, by the harmonic series $\sum_{n=1}^\infty 1/n\,$)

In view of (c), it is helpful to have a good idea before you start whether the ratio test is likely to be helpful or not. My rule of thumb here is that if you see things like factorials, or constants raised to the power $n$, then the Ratio Test is likely to be helpful. Examples include sequences like $x_n= n^2 2^n/3^n$ (giving $L=2/3 < 1$), $x_n=10^n/n!$ (giving $L=0 < 1$) or $x_n= 2^n n!/(2n)!$ (again giving $L=0 < 1$). In all three cases, (b) applies. For something like $x_n= 4^n/(3^n n^3)$ we get $L=4/3>1$ and (a) applies instead.

More importantly, if $x_n$ is just a rational function of $n$ (that is, $p(n)/q(n)$ for some polynomials $p$ and $q$, avoiding division by $0$) then you should never use the Ratio Test, because (assuming you don't end up with division by $0$) the limiting ratio will always be $1$ in that case. So you are stuck in the inconclusive (and so completely unhelpful) case (c).

Here are some easy examples of sequences $(x_n)_{n\in\N}$ for you to think about, where you can check easily that the Ratio Test is inconclusive:

(i) $x_n=n$;

(ii) $x_n=1/n$;

(iii) $x_n=1/n^2$.

More generally, let $k\in \mathbb{Z}$ be some (integer) constant, and consider the sequence where $x_n=n^k$. Again the limiting ratio is $1$. Actually $k$ doesn't have to be an integer: for any real constant $\alpha$, the Ratio Test is inconclusive for the sequence $(n^{\alpha})_{n\in\N}$ (but if $\alpha$ is not an integer, then $n^\alpha$ isn't a rational function of $n$).

Here are some slightly less obvious examples to think about, where the Ratio Test is inconclusive, and you need a different method to investigate the sequence $(x_n)$ and the series $\sum_{n=1}^\infty x_n$:

(iv) $\displaystyle x_n=\frac{2n^4 + 3n^3 +5}{5n^4+7n^2+2}$;

(v) $\displaystyle x_n=\frac{2n^4 + 3n^3 +5}{5n^5+7n^2+2}$;

(vi) $\displaystyle x_n=\frac{2n^4 + 3n^3 +5}{5n^6+7n^2+2}$.

Exercises

(1) For examples (iv) to (vi), determine whether or not the sequence $(x_n)$ converges, and, if it converge, determine the limit.

(2) For examples (iv) to (vi), determine whether or not the series $\displaystyle\sum_{n=1}^\infty x_n$ converges. (You are not expected to determine the value of the sum.)

(3) Prove my claim above about general rational functions of $n$. In other words,  let $p$ and $q$ be polynomials (with real coefficients) and suppose that $p$ and $q$ have no zeros in $\N$ (to avoid division by $0$, but not actually a major issue in interesting cases). Set $x_n=p(n)/q(n)$. Prove that

\[\frac{|x_{n+1}|}{|x_n|} \to 1 \text{ as } n\to\infty\,.\]

Hint: First show that $x_n= n^k y_n$ for some (integer) constant $k$ and some sequence $(y_n)$ such that $(y_n)$ converges to a non-zero real number.