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11: 3.7 Ordinary Differential Equations

… ►The method consists of a set of rules each of which is equivalent to a truncated Taylor-series expansion, but the rules avoid the need for analytic differentiations of the differential equation. …

12: 10.40 Asymptotic Expansions for Large Argument

… ►In the expansion (10.40.2) assume that $z>0$ and the sum is truncated when $k=\mathrm{\ell }-1$. …

13: 18.2 General Orthogonal Polynomials

… ►When the Jacobi matrix in (18.2.11_9) is truncated to an $n\times n$ matrix …

14: 19.36 Methods of Computation

… ►If the iteration of (19.36.6) and (19.36.12) is stopped when ($M$ and $T$ being approximated by $a_s$ and $t_s$, and the infinite series being truncated), then the relative error in $R_F$ and $R_G$ is less than $r$ if we neglect terms of order $r^2$. …

15: 2.10 Sums and Sequences

… ►In both expansions the remainder term is bounded in absolute value by the first neglected term in the sum, and has the same sign, provided that in the case of (2.10.7), truncation takes place at $s=2m-1$, where $m$ is any positive integer satisfying $m\ge \frac{1}{2}(\alpha +1)$. …

16: 9.7 Asymptotic Expansions

… ►In (9.7.5) and (9.7.6) the $n$th error term, that is, the error on truncating the expansion at $n$ terms, is bounded in magnitude by the first neglected term and has the same sign, provided that the following term is of opposite sign, that is, if $n\ge 0$ for (9.7.5) and $n\ge 1$ for (9.7.6). …

17: 18.39 Applications in the Physical Sciences

… ►For either sign of $Z$, and $s$ chosen such that $n+l+1+(2Z/s)>0$, $n=0,1,2,\mathrm{\dots }$, truncation of the basis to $N$ terms, with $x_i^N\in [-1,1]$, the discrete eigenvectors are the orthonormal $L^2$ functions …