Acta Scientiarum Universitatis Pekiniensis (Naturalum)

68 Gy te ee ae 1958 46

a

nP

> os a (7b)

nlognlog log n---log’ n+ [1 (n) ]?

In this connexion the analogue of the harmonic series ¥ is

which is also convergent for p>1 and divergent for p<.

We may of course make other choice of the mn’ in Theorem 7, and make (according to Theorem 2) the corresponding change of the ratio _ in (7C) and (7D). We may also use (7b) instead of (7a).

. 5. By changing the /(m) in the last factor of the denominator of the n-th term of series (7b) into the sum of the first n terms of the series for p=0, and . by expanding the ratio of each pair of consecutive terms of the resulted series, we find that a series of positive terms Sa, is convergent if

all 1 alt 1

Ona n nlogn nlognloglogen nlognlog log n---log’™n

— : — 10 gc! nlognlog log n---log’n «1 (n) ( ny > ( )

where p>1 is independent of m; and it is divergent if

al 1 1 A!

Gy, = 2i— “ mloo loom. lool: nlognlog log n---log (On

Gy n nlogn nlognlogloen

is z Zo (8D’)

n log nlog log n---log™ nel (n)

We write R?(/(n)) for the right member of. inequality (8C’) and consider the extreme values of R?(h) for h=1,2,---, (mn); and find that R*(h) attains the minimum at h=l(n) and R?(k), when p>1, attains the maximum at h=/' (m) > where (’(2) is the integer such that ;

log” MnsU' (n) Slog! In =1. We have Lemma 2. The series

SU, (p) = : (8a)

nlognloglogn---log’ n+ [U' (n) J?

as convergent for p>2 and divergent for p< 2.

Comparing with series (8a) we find that the series

oe —_ = LT Cn) nlogn log log n---log’ Cm

n YA, a : TI Lice? @(m=DI MFO -L,G)? m=3

is conyergent for p>1 and divergent for p<. For this series we have A il al ill 1

= = = sari : Ay1 n nlogn mnlogn log logn nlogn log logn---log’ (tn