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