Calculating the Sum of Expressions Containing Infinity by Definite Integral

Pre-knowledge

• The concept of the Riemann integral
• Newton-Leibniz formula

introduce

According to the concept of definite integral, we can get

$${\int }_a^{b}f(x)dx=\underset{n\to +\infty }{\lim }\frac{1}{n}\sum _{i=1}^{n}f(a+\frac{b-a}{n}i)$$

Then, for parts of the form

$$\underset{n\to +\infty }{\lim }\frac{1}{n}\sum _{i=1}^n{a}_i$$

This kind of formula can be obtained by definite integral.

example

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

Solution:
\qquad原式$=\underset{n\to +\infty }{\lim }\frac{1}{n}\sum _{i=1}^n\frac{1}{1+(\frac{i}{n}{)}^2}={\int }_0^1\frac{1}{1+x^2}dx=\arctan x{|}_0^1=\frac{}{4}$