178
above is less than or equal to
CIP1I + ... + IPNJ)M
IP1 + · · · + Pnl
Chapter 4But the assumptions also imply that IP1 + · · · + Pnl --:+ oo. Hence there is
N' such that for n > N' the inequality
(IP1! + ... + IPNJ)M < :_
IP1 + · · · + Pnl 2
holds. Thus for n > max(N, N') we have lwnl < E, which shows that
Wn -+ 0. Next, suppose that L -:f. 0. Then z~ = Zn - L is a null sequence,
and by what has been shown above,
lim P1(z1-L)+···+Pn(zn-L) =O
n-+oo Pl + · · · + P·n
or
lim Pl Z1 + · · · + PnZn _ L = O
n-+oo Pl + · · · + Pn
i.e., limn-+oo Wn = L.
(16) In property 15, let PnZn = Zn. Then
Zn- =Zn-+ L
Pn
and this implies that
Z1 + ... + Zn = Sn -+ L
P1 + · · · + Pn Pn
Remark This property also holds if L = +oo. Stolz's rule for sequences
is analogous to L'Hopital's rule for functions.
Examples
1. If Zn = in^2 /(n^2 + 1), we have
1. lm Zn= l" Im ( i = i
n-+oo n-+oo 1 + 1/n2)2. If Zn = [(1 - i)/3]n, then limn-+oo Zn = 0 since
(y'2)n
lzn - OI = lznl = 3 -+ 0 as n-+ oo
3. The sequence defined by Zn = emri/^2 has no limit (it is oscillating).
The first few terms of the sequence are i, ....,.1, -i, 1, i, ....
- If Zn= (l+ir the sequence diverges to oo. Note that lznl = (y'2)n-+
oo as n -+ oo.