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Mathematical Expression Editor

Use the Ratio Test to determine whether an infinite series converges or diverges.

1 Ratio Test

In this section we will determine whether an infinite series converges or diverges
using the Ratio Test. The idea behind this test is to determine if the given
series is behaving similarly to a geometric series. For a geometric series, the
ratio of successive terms is constant, i.e., Furthermore, a geometric series
converges if this ratio is between and . For a general series, the ratio of
successive terms is not constant, but if the limit of the ratio of successive
terms approaches a value, , between and , then the series behaves like a
geometric series with ratio equal to . We state this precisely in the following
theorem.

Ratio Test For the series let The series converges (absolutely) if and diverges if . If
then no conclusion can be made.

For any -series, and so the ratio test gives no conclusion about -series.

example 1 Determine whether the series converges or diverges:

By the Ratio Test, Since , the series converges.

(problem 1a) Determine whether the series converges or diverges: By the ratio test,
this series convergesdiverges

(problem 1b) Determine whether the series converges or diverges: By the ratio test,
this series convergesdiverges

(problem 1c) Determine whether the series converges or diverges: By the ratio test,
this series convergesdivergesno conclusion

example 2 Determine whether the series converges or diverges:

By the Ratio Test, Since , the series converges.

(problem 2a) Determine whether the series converges or diverges: By the ratio test,
this series convergesdiverges

(problem 2b) Determine whether the series converges or diverges: By the ratio test,
this series convergesdiverges

example 3 Determine whether the series converges or diverges:

By the Ratio Test, Since , the series converges.

(problem 3a) Determine whether the series converges or diverges: By the ratio test,
this series convergesdiverges

(problem 3b) Determine whether the series converges or diverges: By the ratio test,
this series convergesdiverges

example 4 Determine whether the series converges or diverges:

By the Ratio Test, Since , the series converges.

(problem 4) Determine whether the series converges or diverges: By the ratio test,
this series convergesdiverges

example 5 Determine whether the series converges or diverges:

By the Ratio Test,

Since , the series converges.

We used the famous limit:

(problem 5) Determine whether the series converges or diverges: By the ratio test,
this series convergesdiverges

2 Video Lessons

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