The Relations Between Borel's and Cesàro's Methods of Summation

Abstract

In these general explanations (as in the phrase " Theory of Divergent Series") it is convenient to use divergent as meaning simply non-convergent.In detailed work it is essential to distinguish between divergent and oscillatory series.+ Hardy, Proc.London Math.Soc, Ser. 2, Vol.8, p. 301; Littlewood, ibid., Vol. 9, p. 434.The results of the first of these papers may be deduced as corollaries from those of the second: they have been extended, in a somewhat different direction, by Landau (Prac Matematyczno-Fizycznych, t.21, p. 97), who shows that, when a,, is real, it is enough to suppose nan < K or na n > -K, and makes an interesting application of the result to the Theory of Prime Numbers.% If 2a n x H ->s, as se->l, and na,,->0 as 7t-»oo, then 2a,, is convergent (to sum s).For a proof see Littlewood, I.e.supra, and Bromwich, Infinite Series, p. 251.* Bromwich, Infinite Series, p. 297.f We need hardly point out that s* does not stand for a power of s,,

Locations

• Proceedings of the London Mathematical Society - View
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