Calculus: Early Transcendentals (2nd Edition)

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

Chapter 8 - Sequences and Infinite Series - 8.3 Infinite Series - 8.3 Exercises: 19

Answer

$$\sum_{k=0}^\infty \left(\frac{1}{4} \right)^k=\frac{4}{3} $$

Work Step by Step

To determine what an infinite geometric series converges to we use the following formula: $$\sum_{k=0}^\infty ar^k=\frac{a}{1-r} $$ so we must identify $a$ and $r$ for our series $$\sum_{k=0}^\infty \left(\frac{1}{4} \right)^k $$ $a=1$ and $r=\frac{1}{4}$ we plug this into above formula: $$\sum_{k=0}^\infty \left(\frac{1}{4} \right)^k=\frac{1}{1-\frac{1}{4}}=\frac{4}{3} $$
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