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 anon-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.