Cesàro Sums of Some Unit Sequences

Consider the series with terms

{1,-1,1,-1,1,…}

, namely

∞

∑

k=1

k+1

(-1)

. The series does not converge because its sequence of partial sums,

s

n

=

n

∑

k=1

k

(-1)

={1,0,1,0,1,0,1,…}

, does not have a limit as

n∞

. However, this sequence



s

1

,

s

1

+

s

2

,

s

1

+

s

2

+

s

3

,…=1,

1

2

,

2

3

,

1

2

,

3

5

,…

has the limit

1

2

, which is known as its Cesàro sum.

Experiment with different sign patterns to see the effect on the averages of the partial sums of the terms.