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Mathematics
Mathematics
Problem
For
∣
x
∣
<
1
|x|<1
∣
x
∣
<
1
, let
h
(
x
)
=
∑
n
=
1
∞
(
−
1
)
n
+
1
x
n
n
h(x)=\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^{n}}{n}
h
(
x
)
=
n
=
1
∑
∞
(
−
1
)
n
+
1
n
x
n
. Determine the smallest positive integer
N
N
N
such that the magnitude of the term
(
−
1
)
N
+
2
(
0.5
)
N
+
1
N
+
1
\displaystyle(-1)^{N+2}\frac{(0.5)^{N+1}}{N+1}
(
−
1
)
N
+
2
N
+
1
(
0.5
)
N
+
1
is less than
10
−
4
10^{-4}
1
0
−
4
, and state that value of
N
N
N
.
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