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Small Intervals where the Partial Sums of a Series Fail to Alternate

u
1.1
m
60
Consider the variant
S(u)=
n
Σ
n=1
(-1)
(
(1/n)
n
-u)
of the series
S(0)=
n
Σ
n=1
(-1)
(1/n)
n
, where
u
is a non-negative parameter. For
u1
, the partial sums
S
N
(u)=
n
N
Σ
n=1
(-1)
(
(1/n)
n
-u)
of
S(u)
alternate as
N
gets large. However, for values of
u
in a certain range, there will be a small interval where the sequence of partial sums fails to alternate. That is, there is an
m
such that
S
m
(u)<
S
m+1
(u)<
S
m+2
(u)
or the reverse,
S
m
(u)>
S
m+1
(u)>
S
m+2
(u)
. It might be an interesting exercise to understand this visually paradoxical phenomenon (see Details for an outline).
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