106 
SERIES 
Example 2. 
Неге 
А ~Х , 1+ 1 + ... + 1\ X n *e ff n 
Qi 
(l+ï+ ••• + !) 
п^е^ 2 п 
log^— 
a n ïi _ — log п + р log п+Н п 
Ipt 
Ipi 
= ^— ! И- ~ 1 + - ! = /X 00 by 81, 6). 
l 2 n I log п J 
Hence A is convergent for р> 0 and divergent for /¿<0. No 
test for p — 0. 
But for p = 0, 
log 
n a n nlpi _ H n — l x n — l 2 n 
V2 — 
£ 
CO 
^0 
*г п 
+ 
51, 
1 
rH 
En 
l 
Ipi 
Ipl. 
= -QO, 
since l 2 n > l s n. Thus A is divergent for p = 0. 
94. A very general criterion is due to Kummer, viz.: 
Let A = a x 4- a 2 + •■•be a positive term series. Let k v k 2 , ••• be a 
set of positive numbers chosen at pleasure. A is convergent, if for 
some constant k > 0. 
*Wi>* » = 1,2,- 
«»+1 
A ¿s divergent if 
R =l+r+ 
к л k„ 
is divergent and 
П "2 
K„< 0 
n = 1, 2, 
For on the first hypothesis 
^ C&i^i ^2^2) 
<*3 < - (k 2 a 2 &3Я3) 
Ы)-