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Abel summability by and Cesaro summability by the Cesaro means of the partial sums
Definition
For a real series , write its inclusive partial sums as
and their Cesaro means as
The series is Cesaro summable to if (The Cesaro means and -summability).
It is Abel summable to if the power series
converges for every and in the one-sided sense of The - limit of at a limit point of . These are summability methods for the zero-indexed series of Series, partial sums, convergence and the sum, divergence, and the tail series; they do not assert ordinary convergence.
Depends on
- Series, partial sums, convergence and the sum, divergence, and the tail series
- The Cesaro means $\sigma_n = (x_0 + \dots + x_n)/(n+1)$ and $(C,1)$-summability
- The $\varepsilon$-$\delta$ limit $\lim_{x \to c} f(x) = L$ of $f : A \to \mathbb{R}$ at a limit point $c$ of $A$
- A real power series about a centre, its interval of convergence, and its radius in $[0,+\infty]$
Used by
- Every convergent real series is Cesaro summable and Abel summable to its ordinary sum Corollary
- 1-2+3-4+⋯ is Abel summable to 1/4 but is not Cesaro summable Counterexample
- Grandi's series is Abel summable to 1/2 but its partial sums do not converge Counterexample
- FALSE: Abel summability alone implies ordinary convergence of a series False statement
- For 0<x<1, the Abel transform of a series is (1-x)²∑_n≥0(n+1)σₙxⁿ, where σₙ are the Cesaro means of its partial sums Lemma
- Abel's limit theorem: if a real series converges to s, then its power series tends to s as x↑1 Theorem
- Frobenius' theorem: Cesaro summability of a real series implies Abel summability to the same value Theorem
- Tauber's theorem: an Abel-summable series with ι(n+1)aₙ→0 converges ordinarily to its Abel sum Theorem
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Sources
- Abel summability, Encyclopedia of Mathematics (standard reference, not scraped)
- Cesàro summation, Encyclopedia of Mathematics (standard reference, not scraped)
- Cesàro summation methods, Encyclopedia of Mathematics (standard reference, not scraped)
- S. Semmes, Rice Math 322 notes (standard reference, not scraped)