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Mathematics
Problem
The function
f
(
x
)
=
e
−
x
f(x)=e^{-x}
f
(
x
)
=
e
−
x
is continuous, positive, and decreasing for
x
≥
1
x\ge 1
x
≥
1
. Given that
∫
1
∞
e
−
x
d
x
=
e
−
1
\displaystyle\int_{1}^{\infty} e^{-x}\,dx = e^{-1}
∫
1
∞
e
−
x
d
x
=
e
−
1
, which of the following intervals must contain the value of
∑
n
=
1
∞
e
−
n
\displaystyle\sum_{n=1}^{\infty} e^{-n}
n
=
1
∑
∞
e
−
n
? a)
(
0
,
e
−
1
)
(0,\,e^{-1})
(
0
,
e
−
1
)
b)
(
2
e
−
1
,
3
e
−
1
)
(2e^{-1},\,3e^{-1})
(
2
e
−
1
,
3
e
−
1
)
c)
(
e
−
1
,
2
e
−
1
)
(e^{-1},\,2e^{-1})
(
e
−
1
,
2
e
−
1
)
d)
(
e
−
1
,
e
−
1
+
e
−
2
)
(e^{-1},\,e^{-1}+e^{-2})
(
e
−
1
,
e
−
1
+
e
−
2
)
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