Question

Determine whether the series ${\int }_1^{\infty }{e}^{-5x}$ converges or diverges.

• A. converges to $\frac{1}{5}$
• B. converges to $-\frac{1}{5}$
• C. always diverges
• D. none of these

Answer 1

Answer: (A)

converges to $\frac{1}{5}$

Given : ${\int }_1^{\infty }{e}^{-5x}dx$

$\stackrel{lim}{c\to \infty }{\int }_1^{\infty }{e}^{-5x}dx=\stackrel{lim}{c\to \infty }{\left[\frac{e^{-5x}}{-5}\right]}_1^c=\stackrel{lim}{c\to \infty }[\frac{e^{-5c}}{-5}-\frac{e^{-5}}{-5}]{\int }_1^{\infty }{e}^{-5x}dx=\frac{1}{5e^5}$

Hence the series converges.