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Acceleration of convergence of some infinite sequences {An} whose asymptotic expansions involve fractional powers of n via the \({\tilde {d}}^{(m)}\) transformation

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Abstract

In this paper, we discuss the application of the author’s \(\tilde {d}^{(m)}\) transformation to accelerate the convergence of infinite series \({\sum }^{\infty }_{n=1}a_n\) when the terms an have asymptotic expansions that can be expressed in the form

$$ a_n\sim(n!)^{s/m}\exp\left[\sum\limits^{m}_{i=0}q_in^{i/m}\right]\sum\limits^\infty_{i=0}w_i n^{\gamma-i/m}\quad\text{as } n\to\infty,\quad s\ \text{integer.} $$

We discuss the implementation of the \(\tilde {d}^{(m)}\) transformation via the recursive W-algorithm of the author. We show how to apply this transformation and how to assess in a reliable way the accuracies of the approximations it produces, whether the series converge or they diverge. We classify the different cases that exhibit unique numerical stability issues in floating-point arithmetic. We show that the \(\tilde {d}^{(m)}\) transformation can also be used efficiently to accelerate the convergence of infinite products \({\prod }^{\infty }_{n=1}(1+v_n)\), where \(v_n\sim {\sum }^{\infty }_{i=0}e_in^{-t/m-i/m}\) as \(n\to \infty \), t ≥ m + 1 an integer. Finally, we give several numerical examples that attest the high efficiency of the \(\tilde {d}^{(m)}\) transformation for the different cases.

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Notes

  1. Clearly, Q(n) ≡ 0 takes place when 𝜃i = 0, i = 0, 1,…,m − 1.

  2. The convergence of infinite series \({\sum }^{\infty }_{n=1}a_{n}\) with \(\{a_{n}\}\in \textbf {b}^{(m)}\), m being arbitrary, can be accelerated efficiently by using the d(m) transformation of Levin and Sidi [10], which can be implemented very economically via the recursive W(m)-algorithm of Ford and Sidi [8]. All this is studied in detail also in Sidi [18, Chapters 6 and 7].

  3. Note that most of the methods mentioned above suffer from lack of numerical stability when applied to infinite series \({\sum }^{\infty }_{n=1}a_{n}\) with an behaving as in (1.5). In addition, there is no reliable way to assess the floating-point accuracies of the approximations they produce.

  4. Clearly, if \(\alpha \in \tilde {\mathbf {A}}^{(\gamma ,m)}_{0}\), then α(x) = xγβ(x), where \(\upbeta \in \tilde {\mathbf {A}}^{(0,m)}_{0}\).

  5. Thus, \(\{a_{n}\}\not \in \tilde {\mathbf {b}}^{(m)}\) if c(n) = an+ 1/an is in \(\tilde {\mathbf {A}}^{(0,m)}_{0}\) and has an asymptotic expansion of the form \(c(n)\sim 1+{\sum }^{\infty }_{i=m+1}c_{i} n^{-i/m}\) as \(n\to \infty \).

  6. The choice \(\hat {\sigma }=1\) results in the “universal” formulation of the \(\tilde {d}^{(m)}\) transformation that is applicable in the presence of all \(\{a_{n}\}\in \tilde {\textbf {b}}^{(m)}\) that we mentioned in the paragraph preceding the last in Section 1.

  7. The explanation for this is twofold: (i) Numerical computations show this. (ii) What matters is not so much the exact value of \({\Gamma }^{(j)}_{n}\) in (3.9)–(3.12) and of \({\Lambda }^{(j)}_{n}\) in (3.14)–(3.18), but rather their orders of magnitude, as explained following (3.18) and as many numerical examples show very clearly.

  8. It is clear that (3.12) is useless when {An} diverges.

  9. Thus, \(\lim _{j\to \infty }A^{(j)}_{n}=S\) (i) for all n ≥ 1 if \({\sum }^{\infty }_{k=1}a_{k}\) converges, that is, if Rγ < − 1, and (ii) for n > m(Rγ + 1) if \({\sum }^{\infty }_{k=1}a_{k}\) diverges, that is, if Rγ ≥− 1, in which case, S is the antilimit.

  10. At the present, we do not have a theorem that covers cases with complex γ when GPS is used with noninteger τ in (3.21).

  11. This means that an tends to zero exponentially or behaves at worst like a fixed power of n as \(n\to \infty \).

  12. Note that \(e^{\kappa \theta _{0}}=1\) only when 𝜃0 is purely imaginary and κ|𝜃0| is an integer multiple of 2π.

  13. For the divergent series considered here, we do not even know whether antilimits exist. The approximations \(A^{(0)}_{n}\), n = 0, 1,…, obtained by applying the \(\tilde {d}^{(m)}\) transformation to these series seem definitely to converge, however. Thus, we can safely conclude that \(\lim _{n\to \infty }A^{(0)}_{n}\) are the antilimits of these series, even though we do not know their nature.

  14. See footnote 6.

  15. Recall that the infinite product \({\prod }^{\infty }_{n=1}f_{n}\) is convergent if \(\lim _{n\to \infty }{\prod }^{n}_{k=1}f_{k}\) exists and is finite and nonzero.

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  1. Computer Science Department, Technion - Israel Institute of Technology, 32000, Haifa, Israel

    Avram Sidi

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Sidi, A. Acceleration of convergence of some infinite sequences {An} whose asymptotic expansions involve fractional powers of n via the \({\tilde {d}}^{(m)}\) transformation. Numer Algor 85, 1409–1445 (2020). https://doi.org/10.1007/s11075-019-00870-z

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