Suppose that [math]a_n>0[/math] and [math]b_n>0[/math] for all [math]n\ge N[/math] ([math]N[/math] an integer).[br][list=1][*]If [math]\lim_{n\to\infty}\frac{a_n}{b_n}=c>0[/math], then [math]\sum a_n[/math] and [math]\sum b_n[/math] both converge or both diverge.[/*][*]If [math]\lim_{n\to\infty}\frac{a_n}{b_n}=0[/math] and [math]\sum b_n[/math] converges, then [math]\sum a_n[/math] converges.[/*][*]If [math]\lim_{n\to\infty}\frac{a_n}{b_n}=\infty[/math] and [math]\sum b_n[/math] diverges, then [math]\sum a_n[/math] diverges.[br][/*][/list]

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