Acta Scientiarum Universitatis Pekiniensis (Naturalum)

66 abs ot KR OH 1938 4¢ -

Th = d,

at ds d= C¢;,) ° - (2D)

In the proofs of Theorem 2 and some other theorems in what follows we make use of y

Lemma 1. If 2, %,---, 2, are real numbers of one and the. same stgn and |e, +t,+---+a,_,|<1, then

(1+ 2,) (1+2,) a (1+ 2,) =1+ (isnot) +9 (@,-+----+a,)?, 0<@<1

2. We extend the comparison test and the ratio test as follows.

Theorem 3. In order that a series of positive terms Sa, should be convergent, it is necessary and sufficient that it should be possible to associate every sufficiently large natural number n with a natural number n', less than n and tending to co with n, such that

Dd) Gn<4(G,) +O (30’)

: m =n a ts also necessary and sufficient that 2t should be possible to associate every’ sufficiently

large natural number n with a natural number n, greater than n and increasing with n, such that > Gn, <4 (a,) sine (38C") n'<m<(n+1)" .

In order that the series of positive terms Sa, should be divergent, it is necessary and sufficient that it should be possible to associate every sufficiently large natural number n with a natural nwmber n', less than n and tending to co with n, such that

SS} dn > (1—C,,) ay; (3D’) Mm =n? z it is also necessary and sufficient that it should be possible to associate every sufficiently large natural number n with a natural nwmber n"', greater than n and increasing with n, such that Say AC) a, (3D") n''<m<(n-+1)" 3. We consider a class of series of positive terms Sa, characterized by the

condition that

Oy all n+1 <1+0(—), (4a) Oy, m Suppose that m, is a sequence of natural numbers such that : ";. ome T< lim he lim PIN --oo, (4b) faa % koa Then, for ™%<m<n<n,,,, we have

mr / 1 AT Ny Tn 6,<(14+—— On <3" Om