---
tag: 447CBEA2Z1
kind: definition
name: Nörlund mean
msc: ["40G05"]
defines: X5R7FV9KWH
uses: [8X25Q4XYMN, 0QE8STGJH2, 0NB643K60Q, 0P0DJHBGBB]
sources:
  - url: https://en.wikipedia.org/wiki/Divergent_series
---

## 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|0QE8STGJH2]] 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|0NB643K60Q]] 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|0P0DJHBGBB]].

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.