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. 

What you are looking for in general is called a geometric integral:
$$\prod_a^b f(x)^{dx} = \lim_{\Delta x \to 0}\prod_{i} f(x_i)^{\Delta x}$$
Where the $x_i$ range from $a$ to $b$ with difference $\Delta x$. This can be seen as a multiplicative analog of the Riemann integral. In fact, it can be shown that in general:
$$\prod_a^b f(x)^{dx} = \exp\left(\int_a^b\ln\left(f(x)\right)\,dx\right)$$
by simply writing out the Rienmann sum for $\ln\left(f(x)\right)$. In this particular case, we get:
$$\lim_{n\rightarrow \infty} \left[ \left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)\cdots \left(1+\frac{n}{n}\right) \right]^\frac{1}{n} = \exp\left(\int_0^1\ln(1+x)\,dx\right) = \frac4e$$
