7 
for n =o of sn is infinite, and is at least equivalent ton*—*. For, 
if this were not the case and if s,, were finite for n ==, or 
equivalent to a lower positive power ‘of n, say n°, then the first 
term of the right-hand member would be of order n'~*, whereas 
the terms of the series occurring in that member would be of order 
mi, and their sum therefore at most of order n’—*. This is in 
contradiction to the initial hypothesis that the left-hand member of 
the equality is of order 7’. 
The lemma has thus been proved. A corollary worthy of notice 
is that the series (36) for a< @ diverges in such a way that the 
upper limit of the sum (86’) for mo is exactly equivalent to 
n'-* (in the second part of our proof we found that it was at least 
equivalent to this power of n). For, if that limit were equivalent to 
a higher power of n, say n’—+?, then the limit between convergence 
and divergence of the series 
ES 5 == Qe 
would, by the or Wel just now, be given by B=6@+ Jd, 
and not by S=@. The same consequence may, for the rest, be 
deduced by observing that from the equality (87) it follows that 
for a< 6 the left-hand member is at most equivalent to nf. 
The same lemma as ae above holds for the series of factorials 
Gate 39 
Basaran Ed passe 8) 
as may be proved in ae the same way. Both lemmas are 
moreover a consequence of one another, because the series (36) and 
(39) converge and diverge for the same values of a, at least if « 
has not exactly the value 6, which is the limit between convergence 
and divergence of the series. 
11. We now construct a sequence of coefficients 
ORE Tae mar Ne at Ree Ci) 
whose upper limit for =o is equivalent ‘i Dn wheres the upper 
limit of the sum (35) is equivalent to n°, 6 now being a positive 
number less than 1. It is not difficult to effect this in different 
manners; but we shall moreover try to secure that the second sum 
quantity 
o a 
n 
(2) ht 
en nn m Sm 5 . . . e "3 . . (41) 
0 
