TITLE: Find the limit $ \lim_{n \to \infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^n$
QUESTION [12 upvotes]: For $a,b,c>0$, Find $$ \lim_{n \to \infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}{3}\right)^n$$
how can I find the limit of sequence above? 
Provide me a hint or full solution.
thanks ^^

REPLY [2 votes]: $a^{\frac 1n}=1 +\dfrac{\log a}{n}+O(\frac1{n^2})$
So,$\Big(\dfrac{a^{\frac 1 n}+b^{\frac 1 n}+c^{\frac 1 n}}3\Big)^n=(1+\frac{\log abc}{3n}+O(\frac1{n^2}))^n\to \sqrt[3]{abc}$ This generalizes easily to 4 or more terms also.