# Cesàro Sums of Some Unit Sequences

Cesàro Sums of Some Unit Sequences

Consider the series with terms , namely . The series does not converge because its sequence of partial sums, =={1,0,1,0,1,0,1,…}, does not have a limit as . However, this sequence has the limit , which is known as its Cesàro sum.

{1,-1,1,-1,1,…}

∞

∑

k=1

k+1

(-1)

s

n

n

∑

k=1

k

(-1)

n∞

,+,++,…=1,,,,,…

s

1

s

1

s

2

2

s

1

s

2

s

3

3

1

2

2

3

1

2

3

5

1

2

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