So
.
Since this simplifies to , you can use L'Hospital's Rule to evaluate the limit.
For a, b, c > 0
What is lim (n-->infinity){ [a^(1/n)+b^(1/n)+c^(1/n)]/3}^n equal to ?
The limit converges, as I have tried to similuate the number. The answer must be in terms of a, b, and c.
I tried to expand by binomial series, but got to nowhere useful.
If anyone could help, I would be appreciated. Thank you very much.
An alternative. Expanding in a natural way the function from to the limit is:
Now, use the substitution and the L'Hopital's rule. You'll obtain:
Fernando Revilla