Calculus: Early Transcendentals (2nd Edition)

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

Chapter 8 - Sequences and Infinite Series - 8.2 Sequences - 8.2 Exercises - Page 616: 21

Answer

$\sqrt[4] e$

Work Step by Step

We are given the sequence: $\left\{\sqrt{\left(1+\dfrac{1}{2n}\right)^n}\right\}$ Rewrite the general term of the sequence: $\sqrt{\left(1+\dfrac{1}{2n}\right)^n}=\sqrt{\left(\left(1+\dfrac{1}{2n}\right)^{2n}\right)^{1/2}}=\left(\left(1+\dfrac{1}{2n}\right)^{2n}\right)^{1/4}$ Use the fact that $\lim\limits_{n \to \infty} \left(1+\dfrac{1}{n}\right)^n=e$ to determine the limit of the given sequence: $\lim\limits_{n \to \infty} \left(\left(1+\dfrac{1}{2n}\right)^{2n}\right)^{1/4}=\left(\lim\limits_{n \to \infty} \left(1+\dfrac{1}{2n}\right)^{2n}\right)^{1/4}=e^{1/4}=\sqrt[4] e$
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