Acta Scientiarum Universitatis Pekiniensis (Naturalum)

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1 5 ATE HY he ME 65

n*( Grit Gn Ve 1 ni Gnlsa Oy! )<i- ; 5 ae On Gi, 5 loon Oy! Oy! 4 log nloglogn---logn ry E : ull 19, ' +n (C, = C41) ae logn TN (Ca 7 Cntai) . (10D )

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[i] Pringsheim, A., Math. Ann. 35, 297—394, 1890.

[21] Knopp, K., Theorie und Anwendung der unendlichen Rethen, 3 Aufl., Berlin 1931. [3] Alexiewicz, A., Studia Math. 16, 80—85, 1957.

ON THE CONVERGENCE OF SERIES OF POSITIVE TERMS Leng Sen-ming (Department of Mathematics and Mechanics)

ABSTRACT

We establish some tests for convergence and divergence of an arbitrary series of positive terms Sa,(a,20). For convenience we take a fixed number 0, 0<0<1, and a fixed convergent series of positive terms 3V,.

1. As regards a necessary condition for the convergence of Sia,, Viz. 4,30, we have

Theorem 1. Tet a, and o, be two sequences of non-negative real numbers, where on 70. Then a necessary and sufficient condition for a,—0 ts that it should be posseble to associate every sufficiently large natural number n with a natural number n', less han n and tending to co wrth n, such that

dy, <O (ay) +0y. (1)

Making use of such an index n’ we have also

Thecrem 2. Let every sufficiently large natwral number n be associated with a natural number n', less than n and tending to ce with n. Then in order that a series of positive terms Su, should be convergent, it 13 sufficient that there should te a

convergent series of positive terms dc, such that, for large n,

a (G x ste EEO ee Oe (20)

In order that the serves of posetive terms da, should be dovergent, it %s sufficient

that there should be a divergent serves of posrtve terms Sid, such that, for large n,