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Growth of partial sums of divergent series


Author: R. P. Boas
Journal: Math. Comp. 31 (1977), 257-264
MSC: Primary 65B15
DOI: https://doi.org/10.1090/S0025-5718-1977-0440862-0
MathSciNet review: 0440862
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Abstract: Let $ \Sigma f(n)$ be a divergent series of decreasing positive terms, with partial sums $ {s_n}$, where f decreases sufficiently smoothly; let $ \varphi (x) = \smallint _1^xf(t)dt$ and let $ \psi $ be the inverse of $ \varphi $. Let $ {n_A}$ be the smallest integer n such that $ {s_n} \geqslant A$ but $ {s_{n - 1}} < A(A = 2,3, \ldots )$; let $ \gamma = \lim \{ \Sigma _1^nf(k) - \varphi (n)\} $ be the analog of Euler's constant; let $ m = [\psi (A - \gamma )]$. Call $ \omega $ a Comtet function for $ \Sigma f(n)$ if $ {n_A} = m$ when the fractional part of $ \psi (A - \gamma )$ is less than $ \omega (A)$ and $ {n_A} = m + 1$ when the fractional part of $ \psi (A - \gamma )$ is greater than $ \omega (A)$. It has been conjectured that $ \omega (A) = \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} $ is a Comtet function for $ \Sigma 1/n$. It is shown that in general there is a Comtet function of the form

$\displaystyle \omega (A) = \frac{1}{2} + \frac{1}{{24}}\{ \vert f\prime (m)\vert/f(m)\} (1 + o(1)).$

For $ \Sigma 1/n$ there is a Comtet function of the form $ \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} + 1/(24m) - \{ 1/48{m^2})\} (1 + o(1))$. Some numerical results are presented.

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1977-0440862-0
Article copyright: © Copyright 1977 American Mathematical Society

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