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Mathematics
Problem
Consider the series
e
−
1
=
∑
k
=
0
∞
(
−
1
)
k
k
!
\displaystyle e^{-1}=\sum_{k=0}^{\infty}\dfrac{(-1)^k}{k!}
e
−
1
=
k
=
0
∑
∞
k
!
(
−
1
)
k
. What is the smallest integer
n
n
n
such that using the first
n
n
n
+1 terms (from
k
=
0
k=0
k
=
0
to
k
=
n
k=n
k
=
n
) guarantees an approximation of
e
−
1
e^{-1}
e
−
1
whose error does not exceed
5
×
10
−
5
5\times10^{-5}
5
×
1
0
−
5
?
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