   Chapter 11.6, Problem 22E

Chapter
Section
Textbook Problem

Use the Ratio Test to determine whether the series is convergent or divergent. 2 3 + 2 ⋅ 5 3 ⋅ 5 + 2 ⋅ 5 ⋅ 8 3 ⋅ 5 ⋅ 7 + 2 ⋅ 5 ⋅ 8 ⋅ 11 3 ⋅ 5 ⋅ 7 ⋅ 9 + ⋅ ⋅ ⋅

To determine

Whether the series is convergent or divergent by using the ratio test.

Explanation

1) Concept:

Use the ratio test

2) The ratio test:

(i) If limnan+1an=L<1, then the series n=1an is absolutely convergent. (and therefore convergent).

(ii) If limnan+1an=L>1 or limnan+1an= then the series n=1an is divergent.

(iii) If limnan+1an=1, then the ratio test is inconclusive, that is, no conclusion can be drawn about the convergence or divergence of an.

3) Given:

23+2·53·5+2·5·83·5·7+2·5·8·113·5·7·9+ · · ·

4) Calculation:

Consider. n=1an=n=12·5·8· · ·(3n-1)3·5·7· · ·(2n+1)

Here,an=2·5·8· · ·(3n-1)3·5·7· · ·(2n+1) and an+1=2·5·8· · ·(3n-1)(3n+2)3·5·7· · ·(2n+1)(2n+3)

By the ratio test, consider,

limnan+1an

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