118 
SERIES 
Then M n+X > M n and 
Moreover M n = oo. For 
But D n = oo. 
3. The series 
2 (M n+1 - M n ) = Zd n 
l 
r — 0, 1, 2, ••• 
form an infinite set of divergent series, each series divergent slower 
than any preceding it. l 0 M n = M n . 
log M n+1 - log M n = log (l + 
M n+ ,-M n 
M n 
This proves the theorem for r = 0. Hence as in 102 we find, 
replacing repeatedly M n by log M n , 
Corollary 1. If we take M n = n, we get the series 91, Ex. 2. 
Corollary 2 (Abel). Let D = d x + d%+ ••• be a divergent positive 
term series. Then 
is divergent. 
We take here M n = T) n . 
Corollary 3. Being given a positive term divergent series D, ive 
can construct a series which diverges slower than D. 
For by 101, 3 we may bring D to the form 
2(^+1 M n ). 
Then 1) diverges slower than D.