# 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, , does not have a limit as . However, this sequence has the limit , which is known as its Cesàro sum.

{1,-1,1,-1,1,…}

∑(-1)

∞

k=1

k+1

s=∑(-1)={1,0,1,0,1,0,1,…}

n

n

k=1

k

n∞

s,,,…=1,,,,,…

1

s+s

1

2

2

s+s+s

1

2

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.