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  "body": "\n## Definition\n\nLet $(p_k)_{k \\ge 0}$ be a sequence of real numbers with $p_0 \u003e 0$ and $p_k \\ge 0$ for all $k$, and put $P_n = p_0 + p_1 + \\cdots + p_n$.\nLet $(s_k)_{k \\ge 0}$ be a [[sequence|0QE8STGJH2]] of complex numbers.\nThe *Nörlund means* of $(s_k)$ with respect to $(p_k)$ are\n$$t_n = \\frac{p_n s_0 + p_{n-1} s_1 + \\cdots + p_0 s_n}{P_n} \\quad (n \\ge 0).$$\nThe sequence $(s_k)$ is $(N, p)$ summable to $s$ if $t_n \\to s$.\nA series is $(N, p)$ summable to $s$ if the sequence of its [[partial sums|0NB643K60Q]] is.\n\n## Remarks\n\nExample: 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]].\n\nThe method is regular, that is, it sends every convergent sequence to its limit, if and only if $p_n / P_n \\to 0$.\n\nNon-example of regularity: for $p_k = 2^k$ one has $P_n = 2^{n+1} - 1$ and $p_n / P_n \\to \\frac12$.\nFor the sequence $s = (1, 0, 0, \\dots)$, which converges to $0$, one gets $t_n = \\frac{2^n}{2^{n+1} - 1} \\to \\frac12$.\nSo this Nörlund method is not regular.\nThe condition $p_0 \u003e 0$ is needed so that every $P_n$ is positive and the means are defined.\n",
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