Limit of summation $sumlimits_{k = 1}^n {dfrac{{{2^{frac{k}{n}}}}}{{n + frac{1}{k}}}} $

Find the limit of summation $$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {\dfrac{{{2^{\frac{k}{n}}}}}{{n + \frac{1}{k}}}}.$$

Can I use $$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {\dfrac{{{2^{\frac{k}{n}}}}}{{n + \frac{1}{k}}}} = \mathop {\lim }\limits_{n \to \infty }\sum\limits_{k = 1}^n {\dfrac{{{2^{\frac{k}{n}}}}}{n}}?$$

Thank you for any help.

Answer

Yes you can but it will be a less than or equal inequality instead. This gives a upperbound which is $\displaystyle \int_{0}^1 2^xdx$. To prove that this is the answer, we write $n+ \dfrac{1}{k} < n + 1= n\left(1+\dfrac{1}{n}\right)\implies L \ge \dfrac{\displaystyle \int_{0}^1 2^xdx}{ \displaystyle \lim_{n \to \infty} \left(1+\dfrac{1}{n}\right)} = \displaystyle \int_{0}^1 2^x dx$