TESTS OF CONVERGENCE FOR POSITIVE TERM SERIES 101 
92. 1. One way, as already remarked, to determine whether 
a given positive term series A — a x +a^-\- ••• is convergent or 
divergent is to compare it with some series whose convergence or 
divergence is known. We have found up to the present the 
following standard series S: 
The geometric series 
1 +d + 9 2 + ••• 
(1 
The general harmonic series 
(2 
The logarithmic series 
(3 
(4 
1 
X 
(5 
nLnLnlZn 
We notice that none of these series could be used to determine 
by comparison the convergence or divergence of the series follow 
ing it. 
In fact, let a n , b n denote respectively the nth terms in 1), 2). 
Then for g<l, g > 0, 
J> n _ _ e~” log y ^ 
a n+ 1 n*g n n* 
by I, 464, 
or using the infinitary notation of I, 461, 
K > «»• 
Thus the terms of 2) converge to 0 infinitely slower than the 
terms of 1), so that it is useless to compare 2) with 1) for conver 
gence. Let g > 1. Then 
or 
This shows we cannot compare 2) with 1) for divergence.