On two conjectures concerning the partial sums of the harmonic series
Author:
Stephen M. Zemyan
Journal:
Proc. Amer. Math. Soc. 95 (1985), 8386
MSC:
Primary 40A05
MathSciNet review:
796451
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Abstract: Let denote the th partial sum of the harmonic series. For a given positive integer , there exists a unique integer such that . It has been conjectured that is equal to the integer nearest , where is Euler's constant. We provide an estimate on which suggests that this conjecture may have to be modified. We also propose a conjecture concerning the amount by which and differ from .
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 [1]
 T. M. Apostol, Calculus, Blaisdell, New York, 1962.
 [2]
 R. P. Boas, Partial solution to advanced problem 5989*, Amer. Math. Monthly 83 (1976), 749.
 [3]
 , Partial sums of infinite series, and how they grow, Amer. Math. Monthly 84 (1977), 237258. MR 0440240 (55:13118)
 [4]
 , Growth of partial sums of divergent series, Math. Comp. 31 (1977), 257264. MR 0440862 (55:13730)
 [5]
 R. P. Boas and J. W. Wrench, Jr., Partial sums of the harmonic series, Amer. Math. Monthly 78 (1971), 864870. MR 0289994 (44:7179)
 [6]
 L. Comtet, Problem 5346, Amer. Math. Monthly 74 (1967), 209. MR 1534204
 [7]
 G. Polya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis, vol. 2, SpringerVerlag, Berlin, 1925, pp. 159, 380381; problems VII, pp. 250, 251.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198507964518
PII:
S 00029939(1985)07964518
Article copyright:
© Copyright 1985
American Mathematical Society
