# Maclaurin

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##### 4: 9.17 Methods of Computation
###### §9.17(i) Maclaurin Expansions
Although the Maclaurin-series expansions of §§9.4 and 9.12(vi) converge for all finite values of $z$, they are cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. …
##### 5: 5.7 Series Expansions
###### §5.7(i) Maclaurin and Taylor Series
For 20D numerical values of the coefficients of the Maclaurin series for $\Gamma\left(z+3\right)$ see Luke (1969b, p. 299). …
##### 8: 25.2 Definition and Expansions
###### §25.2(iii) Representations by the Euler–Maclaurin Formula
25.2.8 $\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-s\int_{N% }^{\infty}\frac{x-\left\lfloor x\right\rfloor}{x^{s+1}}\mathrm{d}x,$ $\Re s>0$, $N=1,2,3,\dots$.
25.2.9 $\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-\frac{1}% {2}N^{-s}+\sum_{k=1}^{n}\genfrac{(}{)}{0.0pt}{}{s+2k-2}{2k-1}\frac{B_{2k}}{2k}% N^{1-s-2k}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}\int_{N}^{\infty}\frac{% \widetilde{B}_{2n+1}\left(x\right)}{x^{s+2n+1}}\mathrm{d}x,$ $\Re s>-2n$; $n,N=1,2,3,\dots$.
##### 10: 2.10 Sums and Sequences
###### §2.10(i) Euler–Maclaurin Formula
This is the Euler–Maclaurin formula. … For extensions of the Euler–Maclaurin formula to functions $f(x)$ with singularities at $x=a$ or $x=n$ (or both) see Sidi (2004, 2012b, 2012a). … The asymptotic behavior of entire functions defined by Maclaurin series can be approached by converting the sum into a contour integral by use of the residue theorem and applying the methods of §§2.4 and 2.5. …