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.

Comparison Test

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.

Limit Comparison Test

Suppose that $a_{n}>0$ and Failed to parse (unknown function "\math"): {\displaystyle b_n>0<\math> for each <math>n\in\mathbb{N}} . If Failed to parse (unknown function "\ra"): {\displaystyle \lim_{n\ra\infty}\frac{a_n}{b_n}=L} and $L$ is finite and positive, then $\sum _{n=1}^{\infty }a_{n}$ and $\sum _{n=1}^{\infty }b_{n}$ are either both converges or both diverges.