Question

`lim_(n->oo)(9*9^(1/3)*9^(1/9)*9^(1/27)*......*9^(1/(3^n))` ?

Answer 1

`27`

Explanation:

`prod_(k=0)^oo 9^(1/(3^k)) = 9^(sum_(k=0)^oo 1/3^k)`

and

`sum_(k=0)^oo 1/3^k = 1/(1-1/3) = 3/2` hence

`prod_(k=0)^oo 9^(1/(3^k)) = 9^(3/2) = 27`