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Some Theorems Connected with Maclaurin’s Integral Test

Published online by Cambridge University Press:  03 November 2016

C. T Rajagopal
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Extract

The theorem of J E. Littlewood stated below includes Maclaurin’s integral test and furnishes the basis for a unified theory of convergence criteria for series of positive terms.

Theorem A. If F(x) is positive monotone decreasing and (Dn) is such that

then converges or diverges with .

Type
Research Article
Information
The Mathematical Gazette , Volume 23 , Issue 257 , December 1939 , pp. 456 - 461
Copyright
Copyright © Mathematical Association 1939

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References

page no 456 note * Note on the convergence of series of positive terms, Messenger of Maths., 39 (1910), 191-2.

page no 456 note † For a discussion of this point of view, vide Rajagopal, C. T., Convergence theorems for series of positive terms, Journ. Indian Math. Soc. (New Series), 3 (1938), 118–125 Google Scholar.

page no 457 note * Bromwich, Infinite Series (1926), 76.

page no 457 note † Theorems connected with Maclaurin’s test for the convergence of series, Proc. Lond. Math. Soc. (2), 9 (1911), 126-144, Ths. 1 and 2.

page no 458 note * The divergence of a complex series is defined by Bromwich (op. cit. 233) as the divergence of the sequence of moduli of the partial sums of the series.

page no 458 note † The particular case Dn=n is proved by Bromwich (op. cit. 235-7) by a method which in principle is not very different from the one employed here.

page no 459 note * It is obvious that this series is convergent for k > 1.

page no 460 note * Math. Gazette, 21 (1937), 161-3.

page no 461 note * It may be noticed that if (Dn ) is any strictly increasing divergent sequence, a positive decreasing sequenee, the condition is necessary for the convergence of Σan . From this result we can deduce the divergence of by talking

page no 22 note † On an integral test of Brink, R. W for the convergence of series, Bull. Amer, Math. Soc. 43 (1937), 405–412, Th. 3Google Scholar.