Question

Determine whether the series ${\int }_1^{\infty }\frac{1}{x^2}$ converges or diverges?

• A. always diverges
• B. converges to $1$
• C. converges to $-1$
• D. converges to $2$

Answer 1

Answer: (B)

converges to $1$

Given : ${\int }_0^{\infty }\frac{1}{x^2}dx$

$\stackrel{lim}{c\to \infty }{\int }_1^c\frac{1}{x^2}dx=\stackrel{lim}{c\to \infty }{[-\frac{1}{x}]}_1^c=\stackrel{lim}{c\to \infty }-[\frac{1}{c}-\frac{1}{1}]{\int }_0^{\infty }\frac{1}{x^2}dx=-(0-1)=1$

Hence series converges to $1$