Acta Scientiarum Universitatis Pekiniensis (Naturalum)

ete ener

os 1 3 Se FS TE UA we Be BY he Be HE ~ 69

- Se ee S|, |<+e5, nlognloglogn---log’ (Pn

We thus arrive at

Theorem 8. Let the real number p>1 be independent of n and the integer h=h(n)

be such that 1<log'n<n. Then a series of positive terms Sa, 13 convergent if

a ie So 1 d,, nm nilogn nlognloglogn nlogn log logn---log*1n a cowl +0(C) (86)

nlog nlieg log n---log'n and divergent uf 1 1 il _ 1 Caer nm nilogn 7 log n log log n n log n log log n---log'tn _ 1+1/h nlogn log log n---log'n

a) (C,,) e (8D)

In particular, Da, is convergent if An 1 1 el al

ala a : = a, nn nilogn nlogniloglogn n log n log log n---log'1n

P 1 O ¢ YY

er eee epee CC) (8C,) nlognloglogn---log'n

and divergent if

Gn sy Lod yl uf alt

On —1 mn niogn niegnloglogn n log n log log n---log*“*n

1 ; 9 Sapa peer seem merous (C/o) (8D,) nlog nlog log n---log"n

From this we deduce

Theorem 9. Le 8 be a positive number. La n' be a natural number whach depends on n and has the form

nm =logn+O(1),

Then a series of positive terms Sa, 43 convergent if, for large n,

On —_ 1 Cy! Zz 6 a i

<i +0,—-—C; (9C)

n eet hs Any TU Oy 4 nm nlognleglogn---logtn n

and divergent if, for large n,

ale (9D)

Oy 1 cl ¥ im Oats m

ae in! — 1 2 5 ae a Wy TL Wy! 4 m nlognloglogn---log(n 6. The following lemma gives tests of another character. Lemma 3. Let s, and 1, be two sequences of real numbers such that S_<t, +0 (1) (10) when n tends to ce through an infinite sct of natural numbers, and such that

Sn41— Sinz trys alt (10: )