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

$Aborted

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

u≠1

, 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).