Ratio Test

The ratio test is useful to conclude the divergence/convergence of a Series containing either exponents or factorials.

Definition §

Let $a_n$ be a sequence of real numbers with positive terms only, and ${\sum }_{n=1}^{\infty }{a}_n$ be it’s respective series.

$$L=\underset{n\to \infty }{\lim }\frac{a_{n+1}}{a_n}$$

1. If $L<1$ then the ratio between terms is less than 1 and hence ${\sum }_{n=1}^{\infty }{a}_n$ will eventually converge.
2. If $L>1$ then the ratio between terms is greater than 1 and ${\sum }_{n=1}^{\infty }{a}_n$ diverges
3. If $L=1$ the test is inconclusive and other tests are needed.