# limit comparison test

The following theorem is a powerful test for convergence of series.

###### Theorem 1 (Limit ).

Let $\sum_{n=0}^{\infty}a_{n}$ and $\sum_{n=0}^{\infty}b_{n}$ be two series of positive numbers.

1. 1.

If the limit

 $\lim_{n\to\infty}\frac{a_{n}}{b_{n}}=L$

exists and $L\neq 0$ is a non-zero finite number, then both series $\sum_{n=0}^{\infty}a_{n}$ and $\sum_{n=0}^{\infty}b_{n}$ converge or both diverge.

2. 2.

If $L=0$ and $\sum_{n=0}^{\infty}b_{n}$ converges then $\sum_{n=0}^{\infty}a_{n}$ converges as well. If $L=0$ and $\sum_{n=0}^{\infty}a_{n}$ diverges then $\sum_{n=0}^{\infty}b_{n}$ diverges as well.

3. 3.

Similarly, if the limit is infinite (“$L=\infty$”) and $\sum_{n=0}^{\infty}a_{n}$ converges then $\sum_{n=0}^{\infty}b_{n}$ converges as well. If $L=\infty$ and $\sum_{n=0}^{\infty}b_{n}$ diverges then $\sum_{n=0}^{\infty}a_{n}$ diverges as well.

Title limit comparison test LimitComparisonTest 2013-03-22 15:01:31 2013-03-22 15:01:31 alozano (2414) alozano (2414) 4 alozano (2414) Theorem msc 40-00 DeterminingSeriesConvergence SequenceDeterminingConvergenceOfSeries