Definite Integral Finding the Sum of Expressions Containing Infinity

Pre-knowledge: Find the sum of expressions containing infinity by definite integral

Exercise 1

计算$\underset{n\to +\infty }{\lim }\frac{1^{\frac{1}{3}}+2^{\frac{1}{3}}+\cdots +n^{\frac{1}{3}}}{n^{\frac{4}{3}}}$

Solution:
\qquad原式$=\underset{n\to +\infty }{\lim }\frac{1}{n}\sum _{i=1}^n(\frac{i}{n}{)}^{\frac{1}{3}}={\int }_0^1{x}^{\frac{1}{3}}dx$

$=\frac{3}{4}{x}^{\frac{4}{3}}{|}_0^1=\frac{3}{4}$

Exercise 2

Definition $\underset{n\to +\infty }{\lim }(\frac{1}{n+1}+\frac{1}{n+2}+\cdots +\frac{1}{n+n})$

Solution:
\qquad原式$=\underset{n\to +\infty }{\lim }\frac{1}{n}\sum _{i=1}^n\frac{1}{1+\frac{i}{n}}$

$={\int }_1^2\ln xdx=\ln 2$