Mathematics - Definite Integration Question with Solution | TestHub

${\lim }_{n\to \infty }{\sum }_{k=1}^n\frac{k^{\frac{1}{a}}\left\{n^{a-\frac{1}{a}}+k^{a-\frac{1}{a}}\right\}}{n^{a+1}}$

$$\begin{array}{rl}& =\underset{n\to \infty }{\lim }\sum _{k=1}^n\frac{1}{n}\cdot \left\{{\left(\frac{k}{n}\right)}^{1/a}+{\left(\frac{k}{n}\right)}^a\right\}\\ & ={\int }_0^1\left(x^{1/a}+x^a\right)dx\\ & ={\left\{\frac{x^{(1/a)+1}}{\frac{1}{a}+1}+\frac{x^{a+1}}{a+1}\right\}}_0^1\\ & =\frac{a}{a+1}+\frac{1}{a+1}=1\end{array}$$