5.6 Ratio and root tests  (Page 2/8)

Root test

The approach of the root test    is similar to that of the ratio test. Consider a series ${\displaystyle \sum }_{n=1}^{\infty }{a}_n$ such that $\underset{n\to \infty }{\text{lim}}\sqrt[n]{\left|a_n\right|}=\rho$ for some real number $\rho .$ Then for $N$ sufficiently large, $\left|a_N\right|\approx {\rho }^N.$ Therefore, we can approximate ${\displaystyle \sum _{n=N}^{\infty }|}{a}_n|$ by writing

The expression on the right-hand side is a geometric series. As in the ratio test, the series ${\displaystyle \sum }_{n=1}^{\infty }{a}_n$ converges absolutely if $0\le \rho <1$ and the series diverges if $\rho \ge 1.$ If $\rho =1,$ the test does not provide any information. For example, for any p -series, ${\displaystyle \sum _{n=1}^{\infty }1}\text{/}{n}^p,$ we see that

To evaluate this limit, we use the natural logarithm function. Doing so, we see that

Using L’Hôpital’s rule, it follows that $\text{ln}\rho =0,$ and therefore $\rho =1$ for all $p.$ However, we know that the p -series only converges if $p>1$ and diverges if $p<1.$

The root test is useful for series whose terms involve exponentials. In particular, for a series whose terms $a_n$ satisfy $\left|a_n\right|=b_n^n,$ then $\sqrt[n]{\left|a_n\right|}=b_n$ and we need only evaluate $\underset{n\to \infty }{\text{lim}}{b}_n.$

Choosing a convergence test

At this point, we have a long list of convergence tests. However, not all tests can be used for all series. When given a series, we must determine which test is the best to use. Here is a strategy for finding the best test to apply.