_n=k^

Ratio Test
Examples

Ratio test: Let a_n be a sequence of nonnegative numbers.

• if $\lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n}$ exists and is less than 1, then $\sum a_n$ converges
• if $\lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n}$ does not exist or if it exists and is greater than 1, then $\sum a_n$ diverges.

The ratio test is in some ways much simpler than the comparison test. You don't have to look around for some other series to work with, you just use the terms of the series that you are given.

Let's look at some examples.

1. Determine whether the series $\sum_{n=k}^{\infty}\frac{2^n}{n!}$ coverges or diverges.

Apply the ratio test:

$$\lim_{n \to \infty}\left\vert \frac{\frac{2^{n+1}}{(n+1)!}}{\frac{2^n}{n!}} \right\vert \lim_{n \to \infty}\frac{2}{n+1}$$ = (1)

= 0 < 1 (2)

Since the result of the ratio test is a number less than 1, we get that the series converges. Simple as that!

2. Determine whether the series $\sum_{n=k}^{\infty}\frac{2^n}{n^2}$ coverges or diverges.

$$\lim_{n \to \infty}\left\vert \frac{\frac{2^{n+1}}{(n+1)^2}}{\frac{2^n}{n^2}} \right\vert \lim_{n \to \infty}\frac{2n^2}{n^2 + 2n + 1}$$ = (3)

= 2 > 1. (4)

Since the ratio test returns a result greater than 1, the series must diverge.