The function is continuous

MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

The function

f(x)=e^{-x}

is continuous, positive, and decreasing for

x\ge 1

. Given that

\displaystyle\int_{1}^{\infty} e^{-x}\,dx = e^{-1}

, which of the following intervals must contain the value of

\displaystyle\sum_{n=1}^{\infty} e^{-n}

? (a)

(0,\,e^{-1})

(b)

(2e^{-1},\,3e^{-1})

(c)

(e^{-1},\,2e^{-1})

(d)

(e^{-1},\,e^{-1}+e^{-2})