Acta Scientiarum Universitatis Pekiniensis (Naturalum)

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ay 1 oA FE AS TE FR ee Be BY he Be VE | 167

where h and h’’ are independent of n. From this we deduce (Theorem 4) that

the serves

SS A, : and » Ny, (4) k=1

are both convergent or both divergent, and (Theorem 5) that af Da, catisfres condition

(4a) and is convergent then

a,=0(-).

Te / Also, making us? of Theorem 8, we cbtain Theorem 6. Le Sa, be a series of positive terms catisfying condition (4a), and

la m, be a sequence of natural numbers satisfying condition (4b). Then Sa, 4%

convergent uf

lim 7m —0; (6C) kos Oy

and diveryent if lim nt =o, (6D) k= 50 Cy.

If we apply this theorem to the series

Ser and > i (b=1,2,3,--) (6a)

nlog nlog log n---log*mn (log*n) ”

we see at once that these series are convergent for p>1 and divergent for p<l. 4. We consider a kind of particular choice of the nm’ in Theorem 3.

Theorem 7. Le n' be a natural number which depends on n and has the form

mn =logn+O(1), (7) Then a series of posilive terms Sa, 1s convergent if 1 i at or Gy <6(— ar +C n> ( (C) n and divergent uf Ge ~ C,)y ° (7D)

This test corresponds to the family of series KE)

» q

an 5 (Ta) n log n log log n---log’ Cn

where g>>0 is a parameter and I(n) is the integer such that 1<log”n<e, If we collect the terms according to the equations l(n) =h (h=0,1, 2, 3,---), series

(Ta) takes a form like a geometric series, i.c.

SS 4+0() }-q".

h=0