Chapter 11 - Infinite Sequences and Series - 11.1 Exercises - Page 725: 60

The sequence converges to 5

$a_{n}=\sqrt[n] (3^{n}+5^{n})$ $\lim\limits_{n \to \infty} \sqrt[n] (3^{n}+5^{n})$ $=\lim\limits_{n \to \infty} (3^{n}+5^{n})^{\frac{1}{n}}$ $=\lim\limits_{n \to \infty} (5^{n})^{\frac{1}{n}}((\frac{3}{5})^{n}+1)^{\frac{1}{n}}$ $=\lim\limits_{n \to \infty} 5((\frac{3}{5})^{n}+1)^{\frac{1}{n}}$ $=5\lim\limits_{n \to \infty} ((\frac{3}{5})^{n}+1)^{\frac{1}{n}}$ As $n→\infty$, $(\frac{3}{5})^{n}→0$ Thus $=5\lim\limits_{n \to \infty} ((\frac{3}{5})^{n}+1)^{\frac{1}{n}}=5(0+1)^{0}$ $=5(1)$ $=5$ Therefore, $\lim\limits_{n \to \infty} \sqrt[n] (3^{n}+5^{n})=5$