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

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

$Aborted

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