Limit Comparison Test

Limit comparison test: Let $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty b_n$ be series with $a_n>0$ and $b_n>0$ for all sufficiently large $n$. Suppose the limit

exists and satisfies $0<c<\infty$. Then $\sum a_n$ converges if and only if $\sum b_n$ converges.

This is often used when the comparison test is too rigid, but one can identify the asymptotic size of $a_n$ relative to a known $b_n$ (such as a $p$-series or a geometric series).