Question

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

• A. always converges
• B. always diverges
• C. conditional converges
• D. conditional diverges

Answer 1

Answer: (B)

always diverges

Given : ${\int }_1^{\infty }\frac{1}{\sqrt{x}}dx$

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

Hence it always diverges.