TESTS OF CONVERGENCE FOR POSITIVE TERM SERIES 103 
B converges faster than A, and A converges slower than B. From 
I, 184, we have : 
Let A, B be convergent and 
lim ^=1. 
According as l is 0, y=. 0, go, A converges faster, equally fast, or slower 
than B. 
Returning now to the set of standard series 8, we see that each 
converges (diverges) slower than any preceding series of the set. 
Such a set may therefore appropriately be called a scale of con 
vergent (divergent) series. Thus if we have a decreasing positive 
term series, whose convergence or divergence is to be ascertained, 
we may compare it successively with the scale 8, until we arrive 
at one which converges or diverges equally fast. In practice such 
series may always be found. It is easy, however, to show that there 
exist series which converge or diverge slower than any series 
in the scale 8 or indeed any other scale. 
F0rl6t A, B, «... (2 
be any scale of positive term convergent or divergent series. 
Then, if convergent, 
A?>W>C-'>...; 
«divergent, A n > B n > C n > ••• 
Thus in both cases we are led to a sequence of functions of the 
/iO) >/ a ( w ) >/sO) > - 
type 
Thus to show the existence of a series O which converges (di- 
yerges) slower than any series in 2, we have only to prove the 
theorem: 
3. (Dm Bois Reymond.') In the interval (a, ao) let 
/iO)>/ 2 0*0> — 
denote a set of positive increasing functions ivhich = ao as x = cc. 
A>A>fs> ••• 
Moreover, let