+ ••• 
Passing 
(1 
M >1. 
ily when 
us tind 
PRINGSHEIM’S THEORY 
113 
Hence 
a n+2 | = n ( 1 H \ 1 + 
i \ m/V m. 
ß 
= n(H- 
M + ß 
m 
+ — -fi 
rn mr J\ m m“i 
f + 7 h 
Hence 
n / 
log|a n+2 | = ^ logf 1 + 
i ' 
+ ß—7~1 
m 
+^)=2^ = 
and thus 
oo 
i = lim log | a n+2 1 = 2^- 
Now for a n to = 0 it is necessary that L n = — oo. In 88, Ex. 3, 
we saw this takes place when and only when « + /3 — 7 — 1<0. 
Let us find now when | a n+l \ < | a n |. Now 1) gives 
1 _|_ « + /3 — 7 - 1 + 8, 
n n 
Thus when a + /9 — 7 — 1 < 0, | a„ +2 1 < | a n+l |. Hence in this 
case F is an alternating series. We have thus the important 
theorem: 
The hyper geometric series converges absolutely whevi | x | < 1 and 
diverges when | x | > 1. When x = 1, F converges only when a -f /3 
— 7<0 and then absolutely. When x = —l, F converges only 
when a + /3 — 7 — 1<0, and absolutely if a + /3 — 7 < 0. 
Pringsheim s Theory 
101. 1. In the 35th volume of the Mathematische Annalen 
(1890) Pringsheim has developed a simple and uniform theory oi 
convergence which embraces as special cases all earlier criteria, 
and makes clear their interrelations. We wish' to give a brief 
sketch of .this theory here, referring the reader to his papers for 
more details. 
Let M n denote a positive increasing function of n whose limit 
is + oo for n =00 . Such functions are, for example, g > 0, 
n* , log^w , l»n , Ipilpi ••• l^pilfn 
gs 
I 4*41