Limit and Riemann sums

Evaluate the limit:

$$\lim_{n\rightarrow +\infty}\prod_{r=1}^{n} \left(1+\frac{r}{n}\right)^{\frac{1}{n}}$$

Solution



We denote $L_n$ the given limit and  then :

$$L_n=\prod_{r=1}^{n}\left ( 1+\frac{r}{n} \right )^{1/n} \implies \ln L=\ln \left [ \prod_{r=1}^{n}\left ( 1+\frac{r}{n} \right )^{1/n} \right ]=\frac{1}{n}\sum_{r=1}^{n}\ln \left ( 1+\frac{r}{n} \right )$$

Taking limit at both sides and recognising a Riemman sum at the second side we have that:

$$\ln L_n=\int_{0}^{1}\ln \left ( 1+x \right )\,dx = \ln 4-1$$

Hence:

$$\lim_{n\rightarrow  +\infty}\prod_{r=1}^{n} \left(1+\frac{r}{n}\right)^{\frac{1}{n}}= \frac{4}{e}$$