AP Calculus BC

Ratio Test vs Root Test

Q: Which should I try first?

The ratio test, unless the whole term is visibly raised to the nth power. Factorials in particular collapse beautifully under the ratio test.

The ratio test compares consecutive terms and is the right choice when factorials or products of consecutive integers appear. The root test takes the nth root and is the right choice when the whole term is raised to the nth power. Both are inconclusive when the limit equals one.

Ratio test

Use when: The terms contain factorials, or each term is built from the previous one by multiplying.

Root test

Use when: The entire term is raised to the nth power, so taking an nth root cancels the exponent cleanly.

Side by side

	Ratio test	Root test
Limit taken	$\lim \left\lvert \frac{a_{n+1}}{a_n} \right\rvert$	$\lim \sqrt[n]{\lvert a_n \rvert}$
Best for	Factorials	$n$ th powers
Converges when	$L < 1$	$L < 1$
Inconclusive when	$L = 1$	$L = 1$

Both conclude absolute convergence when the limit is below one, so they handle series with mixed signs directly. Neither needs positive terms, which is why they are the default tools for power series.

When one is inconclusive the other usually is too, because they agree whenever both limits exist. A limit of one means switching to a comparison, the integral test, or the alternating series test.

Where the ratio test really earns its place

Finding the radius of convergence of a power series. Applying it produces an inequality in the variable, and solving that inequality gives the radius.

Frequently asked questions

Which should I try first?

The ratio test, unless the whole term is visibly raised to the nth power. Factorials in particular collapse beautifully under the ratio test.

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