Limit Comparison Test

If $\lim\dfrac{a_n}{b_n}$ is a finite positive number, $\sum a_n$ and $\sum b_n$ converge or diverge TOGETHER.

The Limit Comparison Test avoids needing an exact term-by-term inequality: if $\lim_{n\to\infty}\dfrac{a_n}{b_n}=L$ for some FINITE, POSITIVE $L$ , then $\sum a_n$ and $\sum b_n$ converge or diverge TOGETHER. It's more flexible than the Direct Comparison Test, since matching a series' overall GROWTH RATE (rather than proving a strict inequality) is often much easier.

Examples

For $\sum\frac{n+1}{n^3+2}$ , comparing to $\sum\frac{1}{n^2}$ : $\lim_{n\to\infty}\frac{(n+1)/(n^3+2)}{1/n^2}=1$ , a finite positive number, so both series converge together.