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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 $\displaystyle \frac{a_{n+1}}{an}$ as $n \rightarrow \infty$.

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

\begin{displaymath}\lim_{n \rightarrow \infty} \vert\frac{a_{n+1}}{a_n}\vert=L \end{displaymath}


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

Dina J. Solitro-Rassias