Chapter 11.4, Problem 52E

a.

To determine

To evaluate: The integral ${\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-x}dx}$ and use the integral test to show the series ${\displaystyle \sum _{n=1}^{\infty }{e}^{-n}}$ in convergent.

b.

To determine

To show: The series ${\displaystyle \sum _{n=1}^{\infty }{e}^{-n}}$ is a geometric series and find it sum.

c.

To determine

To conclude: The convergence of ${\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-x}dx}$ implies convergence of the infinite series, its value does not give the sum of the infinite series.