TESTS OF CONVERGENCE FOR POSITIVE TERM SERIES 105 
Thus A is convergent. 
The rest of the theorem follows similarly. 
2. For the positive term series A=a 1 + a 2 + ••■to converge it is 
necessary that, for n — oo, 
lim a n = 0, lim na n = 0, lim najpi = 0, lim najpil^n = 0, ••• 
We have already noted the first two. Suppose now that 
lim nafgt ••• l 8 n> 0. 
Then by I, 338 there exists an mandac> 0, such that 
nafgi • • • l s n > c , n > m, 
or 
Hence A diverges. 
Example 1. 
We saw, 88, Ex. 5, that A is divergent for «<1. For a — 1, 
A is convergent for /3 > 1 and divergent if /3^1, according to 
91, Ex. 2. 
If a > 1, let 
a= a' + a" , a" > 1. 
Then if /3 > 0, 
and .4 is convergent since ^ — is. I f /3 < 0, let 
^ n a 
ß=-ß r , /8' > 0. 
Then 
log^' n 1 
a n = 5-, • —• 
w a w a 
log* 3 ' n < w a ' by I, 463, l ; 
But 
and .4 is convergent since ^ -^77 is.