TITLE: How to compute the limit $\lim_{n \to \infty} (1 + 2^n + 3^n +4^n+5^n)^{1/n}$?
QUESTION [8 upvotes]: How to compute the limit $\lim_{n \to \infty} (1 + 2^n + 3^n +4^n+5^n)^{1/n}$? 
My partial solution: $(1 + 2^n + 3^n +4^n+5^n)^{1/n} \leq (5 \times 5^n)^{1/n}$. Therefore $\lim_{n \to \infty} (1 + 2^n + 3^n +4^n+5^n)^{1/n} \leq \lim_{n \to \infty} 5^{(n+1)/n} = 5$. 
Thank you very much.

REPLY [4 votes]: HINT: $$5^n\le 1+2^n+3^n+4^n+5^n$$