Find $\mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{{3^n} + {5^n}}}$

Question

Find the limit $$\mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{{3^n} + {5^n}}}.$$

Thanks.

What is bigger: $3$ or $5$? Think about it. – Cortizol Jun 25 '13 at 09:15 — Cortizol, Jun 25 '13 at 09:15

score 4 · Answer 1 · answered Jun 25 '13 at 09:25

4

I think simple inequalities give a better intuition of what's going on,

\begin{align} 5=\sqrt[n]{5^n}\leq \sqrt[n]{3^n+5^n} \leq\sqrt[n]{5^n+5^n} = 5\sqrt[n]{2} \to 5 \end{align}

answered Jun 25 '13 at 09:25

davin

3,120

score 1 · Answer 2 · answered Jun 25 '13 at 09:17

1

We have $$ \lim_{n\rightarrow \infty}\sqrt[n]{{{3^n} + {5^n}}}=5\times\lim_{n\rightarrow \infty}\left[1+\left(\frac{3}{5}\right)^n\right]^{\frac{1}{n}}= 5.$$

answered Jun 25 '13 at 09:17

Start wearing purple

54,260

Find $\mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{{3^n} + {5^n}}}$

2 Answers2