AP Calculus BC

nth Term Test vs Ratio Test

Q: Why does the ratio test fail on p-series?

Because the ratio \left( \frac{n}{n+1} \right)^p approaches 1 for every p, so L = 1 and the test is inconclusive whatever the answer turns out to be. Use the p-series rule or the integral test on those.

Q: Does the ratio test prove absolute convergence?

Yes. It is stated with absolute values, so a limit below 1 means \sum \lvert a_n \rvert converges, and the original series converges too. No second test on the absolute values is needed.

The nth term test can only prove divergence: if the terms fail to approach zero the series diverges, and if they do approach zero the test tells you nothing at all. The ratio test can prove either outcome, and it is the tool for factorials, constants raised to the nth power, and power series.

nth term test

Use when: You want a five-second first look, and the terms visibly do not shrink to zero, so divergence is immediate.

Ratio test

Use when: The terms carry factorials or a constant raised to the nth power, or you need the radius of convergence of a power series.

Side by side

	nth term test	Ratio test
Limit computed	$\lim_{n \to \infty} a_n$	$L = \lim_{n \to \infty} \left\lvert \frac{a_{n+1}}{a_n} \right\rvert$
Can prove convergence	Never	Yes, when $L < 1$ , and absolutely
Can prove divergence	Yes, when the limit is nonzero or fails to exist	Yes, when $L > 1$ or the limit is infinite
Inconclusive when	The terms do approach zero	$L = 1$
Best for	A quick screen on any series	Factorials, constants raised to the $n$ th power, power series

The nth term test is a one-way gate. Terms that do not approach zero kill the series outright, but terms that do approach zero prove nothing, because that condition is necessary and nowhere near sufficient. The harmonic series is the permanent reminder: $\frac{1}{n} \to 0$ and the series still diverges.

The ratio test answers in both directions. With $L < 1$ the series converges absolutely, with $L > 1$ or an infinite limit it diverges, and with $L = 1$ it has nothing to say. Factorials are where it shines, since $\frac{(n+1)!}{n!} = n+1$ collapses the ratio to something you can take a limit of by inspection.

Run them in this order

Glance at the terms first, because the nth term test is free and ends the problem when it fires. Only after the terms clear zero do you spend effort on the ratio test, and only reach past it when $L = 1$ sends you to a comparison, the integral test, or the alternating series test.

Frequently asked questions

If the terms go to zero, does the series converge?

No. Terms going to zero is required for convergence but does not deliver it. The harmonic series $\sum \frac{1}{n}$ has terms going to zero and diverges, which is exactly why the nth term test cannot prove convergence.

Why does the ratio test fail on p-series?

Because the ratio $\left( \frac{n}{n+1} \right)^p$ approaches $1$ for every $p$ , so $L = 1$ and the test is inconclusive whatever the answer turns out to be. Use the p-series rule or the integral test on those.

Does the ratio test prove absolute convergence?

Yes. It is stated with absolute values, so a limit below $1$ means $\sum \lvert a_n \rvert$ converges, and the original series converges too. No second test on the absolute values is needed.

In the CED: Unit 10: Infinite Sequences and Series (BC)