120 
SERIES 
It diverges if 
(2 
In the second place, A converges if 
(3 
and diverges if 
d, 
(4 
The tests 1), 2) involve only a single term of the given series 
and the comparison series, while the tests 3), 4) involve two 
terms. With Du Bois Reymond such tests we may call respec 
tively tests of the first and second kinds. And in general any 
relation between p terms 
a n1 a ri+15 a n+p~ 1 
of the given series and p terms of a comparison series, 
c nt G n+\i c n+p-1’ or dn, d n+ i ••• d n+p _i 
which serves as a criterion of convergence or divergence may be 
called a test of the p th kind. 
Let us return now to the tests 1), 2), 3), 4), and suppose we 
are testing A for convergence. If for a certain comparison 
series O 
not always <_Cr , n > m 
it might be due to the fact that c n = 0 too fast. We would then 
take another comparison series C'= which converges slower 
than C. As there always exist series which converge slower than 
any given positive term series, the test 1) must decide the con 
vergence of A if a proper comparison series is found. To find 
such series we employ series which converge slower and slower. 
Similar remarks apply to the other tests. We show now how 
these considerations lead us most naturally to a set of tests which 
contain as special cases those already given. 
106. 1. G-eneral Criterion of the First Kind. The positive term 
series A = a x + a 2 + • • • converges if