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Cesàro Sum

Let s_n=sum_(k=0)^(n)a_k be the partial sums of a series. Define their arithmetic means

sigma_n=1/(n+1)sum_(k=0)^ns_k.

If

lim_(n->infty)sigma_n=S.

then S is called the Cesàro sum of the series, the series is called Cesàro summable to S, and the method of assigning this value is called Cesàro summation. Every convergent series is Cesàro summable to its ordinary sum, while some divergent series are Cesàro summable. For example, 1-1+1-1+... has Cesàro sum 1/2.

See also

Cesàro Convergence, Cesàro Mean, Convergent Series, Divergent Series, Partial Sum

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References

Hardy, G. H. Divergent Series. Providence, RI: American Mathematical Society, 1991.

Cite this as:

Weisstein, Eric W. "Cesàro Sum." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CesaroSum.html

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