The Ratio and Root Tests

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The Ratio Test

The idea behind the ratio and the root tests is to compare the series $\sum \lvert a_n\rvert$ to a geometric series $\sum r^n$ , which we know converges if $\lvert r\rvert<1$ and diverges if $\lvert r\rvert\ge 1$ . This gives us information about whether the series $\sum a_n$ converges absolutely.

The ratio test involves looking at

$\displaystyle{\lim_{n \to \infty} \frac{|a_{n+1}|}{|a_n|}}$ to see how a series behaves in the long run. Notice that, in particular, if

$a_n = r^n$ , then

$\displaystyle\left|\frac{a_{n+1}}{a_n}\right| = r$ .

The Ratio Test: Suppose that

$\displaystyle{\lim_{n\to\infty} \frac{|a_{n+1}|}{|a_n|}} = R.$

If $R < 1$ , then $\sum a_n$ converges absolutely.

If $R > 1$ , then $\sum a_n$ diverges.

If $R=1$ or if $R$ does not exist, then we can't tell. There are examples that converge absolutely, examples that converge conditionally, and examples that diverge.

Roughly speaking, $\sum a_n$ behaves, in the long run, like $\sum R^n$ , except in the borderline case of $R=1$ , where it might converge.