[b]Absolute Convergence[/b][list][*]If [math]\sum\left|a_n\right|[/math] converges, then [math]\sum a_n[/math] converges, and we say that the series [math]\sum a_n[/math] [i]converges absolutely[/i]. [/*][/list][b]Ratio Test[/b][list][*]Let [math]\sum a_n[/math] be any series and suppose that [math]\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=\rho[/math]. Then [b](a)[/b] the series [i]converges absolutely [/i]if [math]\rho<1[/math], [b](b)[/b] the series [i]diverges [/i]if [math]\rho>1[/math], and [b](c)[/b] the test is [i]inconclusive [/i]if [math]\rho=1[/math].[/*][/list][b]Root Test[/b][list][*]Let [math]\sum a_n[/math] be any series and suppose that [math]\lim_{n\to\infty}\sqrt[n]{\left|a_n\right|}=\rho[/math]. Then [b](a)[/b] the series [i]converges absolutely [/i]if [math]\rho<1[/math], [b](b)[/b] the series [i]diverges [/i]if [math]\rho>1[/math], and [b](c)[/b] the test is [i]inconclusive [/i]if [math]\rho=1[/math].[/*][/list]If the expression in the interactive figure is too long to fit on the screen, you can drag it to see if more fully.

[i]Developed for use with Thomas' Calculus, published by Pearson.[/i]