Sequences, series, summability Definition

Nörlund mean

Builds on Summability method Sequence Partial sum Cesàro mean

Definition

Let $(p_k)_{k \ge 0}$ be a sequence of real numbers with $p_0 > 0$ and $p_k \ge 0$ for all $k$ , and put $P_n = p_0 + p_1 + \cdots + p_n$ . Let $(s_k)_{k \ge 0}$ be a sequence of complex numbers. The Nörlund means of $(s_k)$ with respect to $(p_k)$ are $t_n = \frac{p_n s_0 + p_{n-1} s_1 + \cdots + p_0 s_n}{P_n} \quad (n \ge 0).$ The sequence $(s_k)$ is $(N, p)$ summable to $s$ if $t_n \to s$ . A series is $(N, p)$ summable to $s$ if the sequence of its partial sums is.

Remarks

Example: for $p_k = 1$ for all $k$ , one has $P_n = n + 1$ and $t_n = \frac{s_0 + \cdots + s_n}{n+1}$ , the Cesàro mean.

The method is regular, that is, it sends every convergent sequence to its limit, if and only if $p_n / P_n \to 0$ .

Non-example of regularity: for $p_k = 2^k$ one has $P_n = 2^{n+1} - 1$ and $p_n / P_n \to \frac12$ . For the sequence $s = (1, 0, 0, \dots)$ , which converges to $0$ , one gets $t_n = \frac{2^n}{2^{n+1} - 1} \to \frac12$ . So this Nörlund method is not regular. The condition $p_0 > 0$ is needed so that every $P_n$ is positive and the means are defined.

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Sequence definition
Summability method concept
Real sequence definition
Cesàro mean definition
Series definition
Partial sum definition
Nörlund mean definition

Tag	`447CBEA2Z1`
URI	`m2://447CBEA2Z1`
Kind	definition
Writing	W0 stub
MSC	40G05
Defines	Nörlund mean
Uses	Summability method, Sequence, Partial sum, Cesàro mean
Changed	2 November 2026

Nörlund mean

Definition

Remarks

Reading path

Further reading

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