convergents

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1: 17.18 Methods of Computation

… ►Method (1) is applicable within the circles of convergence of the defining series, although it is often cumbersome owing to slowness of convergence and/or severe cancellation. … ►Method (1) can sometimes be improved by application of convergence acceleration procedures; see §3.9. …

2: 3.8 Nonlinear Equations

… ► … ► … ► … ►The rule converges locally and is cubically convergent. … ►The convergence of iterative methods …

3: 3.9 Acceleration of Convergence

§3.9 Acceleration of Convergence

… ► … ►provided that the right-hand side converges. … ►Examples are provided by the following analytic transformations of slowly-convergent series into rapidly convergent ones: … ►For applications to asymptotic expansions, see §2.11(vi), Olver (1997b, pp. 540–543), and Weniger (1989, 2003).

4: T. Mark Dunster

… ►He has received a number of National Science Foundation grants, and has published numerous papers in the areas of uniform asymptotic solutions of differential equations, convergent WKB methods, special functions, quantum mechanics, and scattering theory. …

5: 36.8 Convergent Series Expansions

§36.8 Convergent Series Expansions

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6: 1.9 Calculus of a Complex Variable

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§1.9(v) Infinite Sequences and Series

… ►Absolutely convergent series are also convergent. … ►

§1.9(vi) Power Series

… ► … ►

Dominated Convergence Theorem

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7: 1.12 Continued Fractions

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§1.12(ii) Convergents

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§1.12(iii) Existence of Convergents

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§1.12(v) Convergence

… ►Then the convergents $C_n$ satisfy …The continued fraction converges iff, in addition, …

8: 35.10 Methods of Computation

… ►These algorithms are extremely efficient, converge rapidly even for large values of $m$, and have complexity linear in $m$.

9: 14.13 Trigonometric Expansions

… ►These Fourier series converge absolutely when . If then they converge, but, if $\theta \ne \frac{1}{2}\pi$, they do not converge absolutely. … ►with conditional convergence for each. …

10: 3.10 Continued Fractions

… ► $C_n$ is the $n$th approximant or convergent to $C$. … ►Every convergent, asymptotic, or formal series … ►However, other continued fractions with the same limit may converge in a much larger domain of the complex plane than the fraction given by (3.10.4) and (3.10.5). … ►For the same function $f(z)$, the convergent $C_n$ of the Jacobi fraction (3.10.11) equals the convergent $C_{2n}$ of the Stieltjes fraction (3.10.6). … ►For further information on the preceding algorithms, including convergence in the complex plane and methods for accelerating convergence, see Blanch (1964) and Lorentzen and Waadeland (1992, Chapter 3). …