WOLFRAM|DEMONSTRATIONS PROJECT

Cesàro Sums of Some Unit Sequences

​
number of terms
10
first 10 terms: 1 or -1
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
2
,
s
1
+
s
2
+
s
3
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.