Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 11 - Infinite Sequences and Series - 11.1 Sequences - 11.1 Exercises - Page 745: 60


converges to 5

Work Step by Step

$a_{n}=\sqrt[n] (3^{n}+5^{n})$ $\lim\limits_{n \to \infty} \sqrt[n] (3^{n}+5^{n})$ $=\lim\limits_{n \to \infty} (3^{n}+5^{n})^{\frac{1}{n}}$ $=\lim\limits_{n \to \infty} (5^{n})^{\frac{1}{n}}((\frac{3}{5})^{n}+1)^{\frac{1}{n}}$ $=\lim\limits_{n \to \infty} 5((\frac{3}{5})^{n}+1)^{\frac{1}{n}}$ $=5\lim\limits_{n \to \infty} ((\frac{3}{5})^{n}+1)^{\frac{1}{n}}$ As $nā†’\infty$, $(\frac{3}{5})^{n}ā†’0$ Thus $=5\lim\limits_{n \to \infty} ((\frac{3}{5})^{n}+1)^{\frac{1}{n}}=5(0+1)^{0}$ $=5(1)$ $=5$ Therefore, $\lim\limits_{n \to \infty} \sqrt[n] (3^{n}+5^{n})=5$ The sequence converges to 5
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