Limit Comparison Test

Limit Comparison Test: Let $a_n>0$ and $b_n>0$. If $\lim_{n\to\infty}\frac{a_n}{b_n}=L$ with $0<L<\infty$, then $\sum a_n$ converges if and only if $\sum b_n$ converges.

This test is useful when $a_n$ is asymptotic to a simpler $b_n$.

Proof sketch (optional): For large $n$, $\frac{a_n}{b_n}$ is between, say, $\frac{L}{2}$ and $2L$, yielding inequalities $c_1 b_n\le a_n\le c_2 b_n$ and then applying the comparison test .