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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.