limit of product of $(a_1a_2.\dots a_n)^{\frac{1}{n}}$ How to calculate the following limit 
$$ \lim_{n\rightarrow \infty} \left[ \left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)\cdots \left(1+\frac{n}{n}\right) \right]^\frac{1}{n} .$$
 I was trying this by taking the $\log $ of the product and then limit but I am not getting the answer could anybody please help me. And, also is there any general rule for calculating the limit for  $$ (a_1a_2.\dots a_n)^{\frac{1}{n}}.$$
Thanks. 

Answering your second question as the previous answers all answer your first.

Select a sequence $\{b_n\}$ such that $a_n = e^{b_n}$
Now, instead, we have ${\left({e^{\left(\sum\limits_{i=1}^n b_i \right)}}\right)^{\frac 1 n}} = \left(\prod\limits_{i=1}^n a_i \right)^{\frac 1 n}$
This may allow you to leverage identities involving infinite sums as opposed to infinite products as $n \to \infty$. However, you can't always expect certain properties of the sequence to be preserved by mapping them as powers of $e$.
