ISPA - Ratio Test

Ratio Test

The Ratio Test measures the rate of growth or decrease of a series by examining the ratio equation image indicator $LaTeX: \frac{a_{n+1}}{a_n}$ $\frac{a_{n+1}}{a_n}$ in order to determine whether $LaTeX: \sum^{\infty}_{n=1}a_n$ $\sum^{\infty}_{n=1}a_n$ converges or diverges. The Ratio Test states:

Let equation image indicator $LaTeX: \sum a_n$ $\sum a_n$ be a series with nonzero terms.

i) If equation image indicator $LaTeX: \lim_{n \to \infty }\left|\frac{a_{n+1}}{a_n}\right|<1$ $\lim_{n \to \infty }\left|\frac{a_{n+1}}{a_n}\right|<1$ , then $LaTeX: \sum^{\infty}_{n=1}a_n$ $\sum^{\infty}_{n=1}a_n$ converges absolutely.

ii) If equation image indicator $LaTeX: \lim_{n \to \infty }\left|\frac{a_{n+1}}{a_n}\right|>1$ $\lim_{n \to \infty }\left|\frac{a_{n+1}}{a_n}\right|>1$ or $LaTeX: \lim_{n \to \infty }\left|\frac{a_{n+1}}{a_n}\right|=\infty$ $\lim_{n \to \infty }\left|\frac{a_{n+1}}{a_n}\right|=\infty$ , then $LaTeX: \sum^{\infty}_{n=1}a_n$ $\sum^{\infty}_{n=1}a_n$ diverges.

iii) If equation image indicator $LaTeX: \lim_{n \to \infty }\left|\frac{a_{n+1}}{a_n}\right|=1$ $\lim_{n \to \infty }\left|\frac{a_{n+1}}{a_n}\right|=1$ , then the Ratio test is inconclusive.

View the next presentation that provides examples of how the Ratio Test is used to determine convergence or divergence of an infinite series.

Another perspective on using the Ratio Test to determine convergence is presented below.

Use the interactive applet to explore the Ratio Test by CLICKING HERE. Links to an external site.

Ratio Test Practice

PROBLEM: What does the Ratio Test say about the convergence of the series equation image indicator $LaTeX: \frac{1}{2}+\frac{1}{2}+\frac{1}{4}+\frac{1}{4}+\frac{1}{8}+\frac{1}{8}+...$ $\frac{1}{2}+\frac{1}{2}+\frac{1}{4}+\frac{1}{4}+\frac{1}{8}+\frac{1}{8}+...$ ?

SOLUTION: $LaTeX: \lim_{n \to \infty }|\frac{a_n+1}{a_n}|$ $\lim_{n \to \infty }|\frac{a_n+1}{a_n}|$ does not exist, so the Ratio Test does not apply,

Ratio Test: Even More Problems!

Please review all content and feedback and contact your instructor if additional clarification is needed. Once you feel confident with the material, go to the Quizzes widget and complete the Integral, Comparison, Alternating Series, and Ratio Tests Quiz.

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