Honors III: Unbounded Cesaro summable sequence
September 17th 2021
Here, we give an example of a series whose terms unbounded, but which is Cesàro summable to 0. This
corrects the erroneous example given in recitation, which had σn → ∞ (exercise).
Claim 1. Let
The
P∞
n=1 cn


k
cn = −k


0
n = k 3 for k ∈ N
n = k 3 + 1 for k ∈ N
else
is Cesàro summable to 0.
Proof. The partial sums of the series are
(
k
Sn =
0
n = k 3 for k ∈ N
else
The nth Cesáro mean is given by
S1 + · · · + Sn
n
3 nc3
S1 + S8 + · · · + Sb √
=
n √
1 + 2 + · · · + b 3 nc
=
√
√n
b 3 nc(b 3 nc + 1)
=
n
=O(n−1/3 )
σn =
so σn → 0 as n → ∞.
1