limit comparison test

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

Theorem 1 (Limit ).

Let ${\mathrm{\sum }}_{n\mathrm{=}\mathrm{0}}^{\mathrm{\infty }}{a}_n$ and ${\mathrm{\sum }}_{n\mathrm{=}\mathrm{0}}^{\mathrm{\infty }}{b}_n$ be two series of positive numbers.

1. If the limit

   $$\underset{n\to \mathrm{\infty }}{lim}\frac{a_n}{b_n}=L$$

   exists and $L\ne 0$ is a non-zero finite number, then both series ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n$ and ${\sum }_{n=0}^{\mathrm{\infty }}{b}_n$ converge or both diverge.
   

2. If $L=0$ and ${\sum }_{n=0}^{\mathrm{\infty }}{b}_n$ converges then ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n$ converges as well. If $L=0$ and ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n$ diverges then ${\sum }_{n=0}^{\mathrm{\infty }}{b}_n$ diverges as well.
   

3. Similarly, if the limit is infinite (“$L=\mathrm{\infty }$”) and ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n$ converges then ${\sum }_{n=0}^{\mathrm{\infty }}{b}_n$ converges as well. If $L=\mathrm{\infty }$ and ${\sum }_{n=0}^{\mathrm{\infty }}{b}_n$ diverges then ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n$ diverges as well.

Title	limit comparison test
Canonical name	LimitComparisonTest
Date of creation	2013-03-22 15:01:31
Last modified on	2013-03-22 15:01:31
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	4
Author	alozano (2414)
Entry type	Theorem
Classification	msc 40-00
Related topic	DeterminingSeriesConvergence
Related topic	SequenceDeterminingConvergenceOfSeries