Acta Scientiarum Universitatis Pekiniensis (Naturalum)

70 le RS 1958 4F

for all n. Then

for all n. bl f the t, kave a posetive lower bound, (10a) can be replaced by

Spt > bu f= (10b) Sy ty =

Applying Lemma 3 and Theorem 2, and applying the argument used in the deduction of Theorem 8, we cbtain

Theorem 10. Let the real number p>1 be independent of n und the integer h=h(n) be such thut 1<log'n<n. If Sa, is a series of positive terms such that

~ lim-""= <1, ~ (10)

Na = Oy —4 then uw sufficient condition for the convergence of Sa, ts that, for n sufficiently large,

mt Orta — On Vp 4 ta .. oe Gy Una logn log n log log n lognloglogn---log**n (Ch = Onn): (100)

log nlog log n---log"n

If Sa, ts a serves of positive terms such that

lim Gy > 1 ; : : (10d) Noo Qy—1 then a sufficient condition for the divergence of da, ts that, for n sufficiently lurge, n( at _ Gy \<1 { 1 { il eee | 1 | Gn Gale loen log n log log n lognlog log n---log*n ! / . R pe +13 (0,— Cyst) « (20D)

lognlog logn---log'n | The statements remain true if n'=log n+O(1) and (10C) and (LOD) are replaced by

m( Sat On )- 1 m2 Ay! 44 — Gi! 1+ . o

Gy Gna / logn Uy! (byt aa log n log log n---log’n —W (C,—Cyai) + ie =n (C= Cae (100") a n( Gnta On \- 1 n’?( Gyty4 Ay! a= 8 On M-1/ logn in Ca log n log log n---1eg?n +n? (C,,— Cane at” (OieaCneae (10D") S

If we make us2 of the second part of Lemma 3, we find that there are ana-

logues of these tests for the expression

n?( Ons = a, -1) \ a, Oy —4