How to calculate limit of summation?

Question

How to calculate the following limit?

$$\lim_{n\to +\infty} \left( \dfrac{a^{1/n}}{n+1}+\dfrac{a^{2/n}}{n+1/2}+\dots +\dfrac{a^{n/n}}{n+1/n} \right)$$ where $a>0$.

Can we convert this summation to the integral?

Answer 1

Use Squeeze and Riemann sums.

$$\frac{a^{\frac{k}{n}}}{n+\frac{1}{k}}\leq \frac{a^{\frac{k}{n}}}{n}$$

And $$\sum \frac{a^{\frac{k}{n}}}{n}\to\int _0^1 a^x dx=\frac{a-1}{\ln a}.$$

In the other hand $$\frac{a^{\frac{k}{n}}}{n+1}\leq \frac{a^{\frac{k}{n}}}{n+\frac{1}{k}}$$

And $$\sum \frac{a^{\frac{k}{n}}}{n+1}=\frac{n}{n+1}\sum \frac{a^{\frac{k}{n}}}{n}\to \frac{a-1}{\ln a}.$$