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Cesaro summability implies bounded partial sums

by stripes
Tags: bounded, cesaro, implies, partial, summability, sums
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stripes
#1
Feb6-13, 02:53 AM
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1. The problem statement, all variables and given/known data

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

2. Relevant equations

--

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[itex] c_{n} > 0[/itex] for each [itex] n\geq 0,[/itex] then the series is nondecreasing, which means the partial sums are nondecreasing, so we are looking for an upperbound, not a lowerbound.
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pasmith
#2
Feb6-13, 06:34 AM
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P: 949
Quote Quote by stripes View Post
1. The problem statement, all variables and given/known data

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

2. Relevant equations

--

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


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