Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 8 - Sequences and Infinite Series - 8.1 An Overview - 8.1 Exercises: 73

Answer

a) First Four Terms: $\frac{1}{3},\frac{1}{9},\frac{1}{27},\frac{1}{81}$ b) $\lim\limits_{n \to \infty} S_n = \frac{1}{2}$

Work Step by Step

$$\sum_{k=1}^{\infty} 3^{-k}$$ Part A) $a_1 = 3^{-1} = \frac{1}{3}$ $a_2 = 3^{-2} = \frac{1}{9}$ $a_3 = 3^{-3} = \frac{1}{27}$ $a_4 = 3^{-4} = \frac{1}{81}$ Part B) This is a geometric series with an initial term of $a_1 = \frac{1}{3}$ and factor of $r = \frac{1}{3}$, which satisfies $r < 1$. Since it's a geometric series, a limit does exist. $S_\infty = \frac{a_1}{1 - r} = \frac{\frac{1}{3}}{1-\frac{1}{3}} = \frac{1}{2}$
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