Ratio test

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In mathematics, the ratio test is a test (or "criterion") for the convergence of a series whose terms are real or complex numbers. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test. The ratio test is defined as:

<math>L = \lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|</math>

where

lim denotes the limit as n goes to infinity,
an and an+1 are the nth and (n+1)th terms of an infinite series and
L is a label for the result of the ratio test.

The results of the ratio test show that:

  • if <math> L<1 \!</math> the series converges absolutely, or
  • if <math> L>1 \!</math> the series diverges, or
  • if <math> L=1 \!</math> the test is inconclusive (there exists both convergent and divergent series that satisfy this case).

For example, any series in the form:

<math>\sum_{n=1}^\infty f_n</math>

can be applied to the ratio test.

Contents

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Examples

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Converging

Consider the series:

<math>\sum_{n=1}^\infty\frac{n}{e^n}</math>

Putting this into the ratio test:

\frac{a_{n+1}}{a_n}\right|</math> \frac{\frac{n+1}{e^{n+1}}}{\frac{n}{e^n}}\right|</math>
\frac{n+1}{e^{n+1}}\cdot\frac{e^n}{n}\right|</math>
\frac{n+1}{n}\cdot\frac{e^n}{e^n\cdot e}\right|</math>
(1+\frac{1}{n})\cdot\frac{1}{e}\right|</math>
=<math>1\cdot\frac{1}{e}</math>
=<math>\frac{1}{e} (<1)</math>

Thus the series converges as <math>\frac{1}{e}</math> is less than 1.

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Diverging

Consider the series:

<math>\sum_{n=1}^\infty\frac{e^n}{n}</math>

Putting this into the ratio test:

\frac{a_{n+1}}{a_n}\right|</math> \frac{\frac{e^{n+1}}{n+1}}{\frac{e^n}{n}}\right|</math>
\frac{e^{n+1}}{n+1}\cdot\frac{n}{e^n}\right|</math>
\frac{n}{n+1}\cdot\frac{e^n\cdot e}{e^n}\right|</math>
(1-\frac{1}{n+1})\cdot e\right|</math>
=<math>1\cdot e</math>
=<math>\!\, e (>1)</math>

Thus the series diverges as <math>e</math> is greater than 1.

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Inconclusive

If one has

<math>\lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|=1</math>

it is impossible to deduce from the ratio test if the series converges or diverges.

For example, the series

<math>\sum_{n=1}^\infty 1</math>

diverges, but

<math>\lim_{n\rightarrow\infty}\left|\frac{1}{1}\right| = 1.</math>

On the other hand,

<math>\sum_{n=1}^\infty \frac{1}{n^2} </math>

converges absolutely, but

<math>\lim_{n\rightarrow\infty}\left|\frac{\frac{1}{(n+1)^2}}{\frac{1}{n^2}}\right| = 1.</math>

Finally,

<math>\sum_{n=1}^\infty (-1)^n\frac{1}{n} </math>

converges conditionally but

<math>\lim_{n\rightarrow\infty}\left|\frac{\frac{(-1)^{n+1}}{(n+1)}}{\frac{(-1)^{n}}{n}}\right| = 1.</math>
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L=1 and Raabe's test

As seen in the previous example, the ratio test is inconclusive when the limit of the ratio is 1. An extension of the ratio test due to Raabe sometimes allows one to deal with this case. Raabe's test states that if

<math>\lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|=1</math>

and if a positive number c exists such that

<math>\lim_{n\rightarrow\infty}

\,n\left(\,\left|\frac{a_{n+1}}{a_n}\right|-1\right)=-1-c</math>

then the series will be absolutely convergent. d'Alembert's ratio test and Raabe's test are the first and second theorem in a hierarchy of such theorems due to Augustus De Morgan.

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See also

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References

  • Knopp, Konrad, "Infinite Sequences and Series", Dover publications, Inc., New York, 1956. (§ 3.3, 5.4) ISBN 0486601536

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