Homework Help: Cesaro summability implies bounded partial sums

1. Feb 6, 2013

stripes

1. The problem statement, all variables and given/known data

Suppose $c_{n} > 0$ for each $n\geq 0.$ Prove that if $\sum ^{\infty}_{n=0} c_{n}$ is Cesaro summable, then the partial sums $S_{N}$ are bounded.

2. Relevant equations

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3. The attempt at a solution

I tried contraposition; that was getting me nowhere. I have a few inequalities here and there but they don't tell me anything. I need to show that there exists an upperbound for the partial sums. This means there exists a least upperbound. I need to find that least upperbound. Because$c_{n} > 0$ for each $n\geq 0,$ then the series is nondecreasing, which means the partial sums are nondecreasing, so we are looking for an upperbound, not a lowerbound.

2. Feb 6, 2013

pasmith

That least upper bound, if it exists, is $\lim_{n \to \infty} S_n$, which is the definition of $\sum_{n=0}^{\infty} c_n$ in the traditional sense.