divergence

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11: 2.11 Remainder Terms; Stokes Phenomenon

… ►For divergent expansions the situation is even more difficult. … ►However, regardless whether we can bound the remainder, the accuracy achievable by direct numerical summation of a divergent asymptotic series is always limited. … ►The transformations in §3.9 for summing slowly convergent series can also be very effective when applied to divergent asymptotic series. …

12: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►If $p=q+1$, then (35.8.1) converges absolutely for and diverges for $\Vert 𝐓\Vert >1$. ►If $p>q+1$, then (35.8.1) diverges unless it terminates. …

13: Bibliography F

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Studies on Divergent Series and Summability & The Asymptotic Developments of Functions Defined by Maclaurin Series. Chelsea Publishing Co., New York.

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Notes:

Reprint with corrections of two books published originally in 1916 and 1936.

External Links:

MathReview

Referenced by:

§2.10(iii).

Encodings:

BibTeX

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14: Bibliography H

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Divergent Series. Clarendon Press, Oxford.

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Notes:

Reprinted by the American Mathematical Society in 2000.

External Links:

MathReview (R. P. Agnew), ZentralBlatt

Referenced by:

§1.15(ii), §1.15(viii).

Encodings:

BibTeX

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15: 15.2 Definitions and Analytical Properties

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Diverges when $\mathrm{\Re }\left(c-a-b\right)\le -1$.

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16: 31.11 Expansions in Series of Hypergeometric Functions

… ►Such series diverge for Fuchs–Frobenius solutions. …

17: 2.7 Differential Equations

… ►Hence unless the series (2.7.8) terminate (in which case the corresponding ${\mathrm{\Lambda }}_j$ is zero) they diverge. … ►We cannot take $f=x$ and $g=\ln x$ because $\int g{f}^{-1/2}dx$ would diverge as $x\to +\mathrm{\infty }$. …

18: 2.1 Definitions and Elementary Properties

… ►Let $\sum a_s{x}^{-s}$ be a formal power series (convergent or divergent) and for each positive integer $n$, …

19: Bibliography S

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Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, pp. 1–42.

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External Links:

MathReview (J. Kuntzmann), ZentralBlatt

Referenced by:

§17.18.

Encodings:

BibTeX

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20: 22.3 Graphics

… ►The period diverges logarithmically as $k\to 1-$; see §19.12. …