Direct Comparison Test

Given two series $\sum a_n$ and $\sum b_n$ and assuming there exists a $M>0$ such that $0\le a_n\le b_n$ for all $n>M$

If $\sum b_n$ converges then $\sum a_n$ converges.
If $\sum a_n$ diverges then $\sum b_n$ diverges.

Limit Comparison Test

If $a_n\text{ and }{b}_n$ are positive sequences and the following exists:

• If $L>0$ then $\sum a_n$ converges if and only if $\sum b_n$ converges.
• If $L=\mathrm{\infty }$ and $\sum a_n$ converges, then $\sum b_n$ converges.
• If $L=0$ and $\sum b_n$ converges, then $\sum a_n$ converges.