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 be a divergent series of decreasing positive terms, with partial sums , where *f* decreases sufficiently smoothly; let and let be the inverse of . Let be the smallest integer *n* such that but ; let be the analog of Euler's constant; let . Call a Comtet function for if when the fractional part of is less than and when the fractional part of is greater than . It has been conjectured that is a Comtet function for . It is shown that in general there is a Comtet function of the form

**[1]**M. ABRAMOWITZ & I. A. STEGUN (Editors),*Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables*, Nat. Bur. Standards Appl. Math. Ser., 55, U. S. Government Printing Office, Washington, D.C., 1964. MR**29**#4914.**[2]**R. P. Boas Jr. and J. W. Wrench Jr.,*Partial sums of the harmonic series*, Amer. Math. Monthly**78**(1971), 864–870. MR**0289994**, https://doi.org/10.2307/2316476**[3]**Louis Comtet,*Problems and Solutions: Solutions of Advanced Problems: 5346*, Amer. Math. Monthly**74**(1967), no. 2, 209. MR**1534204****[4]**G. H. HARDY,*Orders of In finitv*. 2nd ed., Cambridge Univ. Press, New York, 1924.**[5]**K. A. Karpov and S. N. Razumovskiĭ,*Tables of the integral logarithm*, Izdat. Akad. Nauk SSSR, Moscow, 1956 (Russian). MR**0085623****[6]**K. KNOPP,*Theory and Application of Infinite Series*, Blackie, London and Glasgow, 1928.**[7]**James Miller and R. P. Hurst,*Simplified calculation of the exponential integral*, Math. Tables Aids Comput.**12**(1958), 187–193. MR**0104348**, https://doi.org/10.1090/S0025-5718-1958-0104348-3**[8]***Tables of Sine, Cosine and Exponential Integrals. Vol. I*, National Bureau of Standards, New York, 1940. Technical Director: Arnold N. Lowan. MR**0003570****[9]**N. J. A. Sloane,*A handbook of integer sequences*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973. MR**0357292**

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DOI:
https://doi.org/10.1090/S0025-5718-1977-0440862-0

Article copyright:
© Copyright 1977
American Mathematical Society