Nörlund Means
Suppose pn is a sequence of positive terms, starting from p0. Suppose also that
If now we transform a sequence s by using p to give weighted means, setting
then the limit of tn as n goes to infinity is an average called the Nörlund mean Np(s).
The Nörlund mean is regular, linear, and stable. Moreover, any two Nörlund means are consistent. The most significant of the Nörlund means are the Cesàro sums. Here, if we define the sequence pk by
then the Cesàro sum Ck is defined by Ck(s) = N(pk)(s). Cesàro sums are Nörlund means if k ≥ 0, and hence are regular, linear, stable, and consistent. C0 is ordinary summation, and C1 is ordinary Cesàro summation. Cesàro sums have the property that if h > k, then Ch is stronger than Ck.
Read more about this topic: Divergent Series
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