The Ratio Test

The Ratio Test for convergence of a series can be thought of as a measurement of how fast the series is increasing or decreasing. This can be found by looking at the ratio $$\frac{a_{n+1}}{an}$$ as $n \rightarrow \infty$.

Given the series $$\sum_{n=0}^{\infty} a_n$$, suppose that

$$\lim_{n \rightarrow \infty} \vert\frac{a_{n+1}}{a_n}\vert=L$$

Then

1. the series converges if $L < 1$,
2. the series diverges if $L>1$,
3. the test is inconclusive if $L=1$.