Comparison test

Comparison test can mean:

• Limit comparison test, a method of testing for the convergence of an infinite series.
• Direct comparison test, a way of deducing the convergence or divergence of an infinite series or an improper integral.

Suppose that 0<a_n\leq b_n for each n\in\mathbb{N}. Then we have:

(i) If $\sum _{n=1}^{\infty }b_{n}$ converges, then $\sum _{n=1}^{\infty }a_{n}$ converges.

(ii) If $\sum _{n=1}^{\infty }a_{n}$ diverges, then $\sum _{n=1}^{\infty }b_{n}$ diverges.