Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 12 - Sequences and Series - 12.4 Infinite Series - 12.4 Exercises - Page 637: 3

Answer

$${\text{the series diverges}}$$

Work Step by Step

$$\eqalign{ & 2 + 6 + 18 + 54 + \cdots \cr & {\text{Sum of a Geometric Series }}\left( {{\text{see page 635}}} \right) \cr & {\text{The infinite geometric series}} \cr & a + ar + a{r^2} + a{r^3} + \cdots \cr & {\text{converges}}{\text{, if }}r{\text{ is in }}\left( { - 1,1} \right),{\text{ to the sum }}\frac{a}{{1 - r}}.{\text{ And diverges if }}r{\text{ is not in }}\left( { - 1,1} \right) \cr & {\text{then this is a geometric series}}{\text{, with }}a = {a_1} = 2 \cr & r = \frac{6}{2} = 3 \cr & {\text{Since }}r{\text{ is not in the interval }}\left( { - 1,1} \right),{\text{ the series diverges}} \cr} $$
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