Cesaro Sums (Real Analysis)

We have defined the convergence of a series,

SUM [a_j] from j=1 to infinity

in terms of the sequence of partial sums associated with that series,

Sn = SUM [a_j] from j=1 to n

We say that the series converges if, and only if, the sequence of its partial sums converges.

The mathematician Ernesto Cesaro introduced another idea to this field, an idea that we call Cesaro summability. We define another sequence,

P_m=(1/m) * SUM[S_n] from n=1 to m

That is, P_m is the arithmetic mean of the first m partial sums. If the sequence {P_m} converges to z, we say that SUM[a_j] from j=1 to infinity is a Cesaro summable and that its Cesaro sum is z.



1) Prove that if a series converges, then it is Cesaro summable and that the series converges to its Cesaro sum.

2) Prove that if a series of positive terms diverges it is not Cesaro summable.




Guys, I'm really stuck on the above problem. Any tip or guidance would be much appreciated. Thanks!

Hey JamesTrack.

Have you looked at the proofs that involve Cauchy-Sequences? Maybe you could adapt those to the question that you need to answer.

It was a while back, but I do recall the proofs of Cauchy-Sequences being used for all kinds of convergence proofs of partial sums.