cesaro summable proof

• April 15th 2010, 03:53 PM
234578
cesaro summable proof
1. Find a sequence which is casaro summable but which is not summable.
2. Suppose that {a(subn)} is a positive sequence and is cesaro summable. Suppose also that the sequence {n*a(subn)} is bounded. Prove that the series $\sum_{n=1}^\infty a(n)$ converges.

Hint: If S(subn) is \sum_{i=1}^\{n} a(subi) and if s(subn) is (1/n)* \sum_{i=1}^\n S(subi) prove that S(su n)-(n/(n+1))s(subn) is bounded.

note: a(subn) cesaro summable means that there exists a limit L=limit n->infinity of the arithmetic average of the partial sums of a(subn)
• April 15th 2010, 07:43 PM
tonio
Quote:

Originally Posted by 234578
1. Find a sequence which is casaro summable but which is not summable.

Try $\{(-1)^n\}$ . Your question (2) is (at least almost) unreadable. Try to type it again correctly using LaTeX as taught in the section "Math Resources: LaTeX Help" (below "University Math Help")

Tonio

2. Suppose that {a(subn)} is a positive sequence and is cesaro summable. Suppose also that the sequence {n*a(subn)} is bounded. Prove that the series $\sum_{n=1}^\infty a(n)$ converges.

Hint: If S(subn) is \sum_{i=1}^\{n} a(subi) and if s(subn) is (1/n)* \sum_{i=1}^\n S(subi) prove that S(su n)-(n/(n+1))s(subn) is bounded.

note: a(subn) cesaro summable means that there exists a limit L=limit n->infinity of the arithmetic average of the partial sums of a(subn)

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