1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences and Series 179

5. If Zn= na/(1 + na), then limn->oo Zn = 0 if a= 0, and limn->oo Zn = 1
   if a =/= 0.

6. The sequence Zn = (-1 )n is oscillating, yet the sequence of the

arithmetic means

Wn= (-1)+(-1)2+···+(-l)n = {0-1

n -

n

according to whether n is even or odd, so that limn->oo Wn = 0. This

shows that the converse of property 15 does not hold.

Remark Cesaro, Holder, Borel, and other authors have generalized the
concept of convergence of a sequence (and of a series) by using lim Wn (or
the limit of some other type of average of the terms of the sequence) as
the generalized limit of {zn}·
For instance, whenever the sequence of the arithmetic means has a limit
L as n ---+ oo, the sequence {zn} is said to be summable to L by Cesaro's
method. In view of a further development of this method [3], it is also
said that {zn} is summable (C, 1) meaning by Cesaro's method of order 1.
Also, it is said that {zn} is summable (H, 1), since the method of arith-
metic means is again the first step in a sequence of methods considered by

0. Holder [7]. Incidentally, the two methods of Cessaro and Holder have

been shown to be equivalent. These methods of summability have been
further extended and modified in various ways (see, e.g., [2], [5], [6], [9],
[10], [11], [12], and [14].)

7. As an application of Stolz's rule, suppose that we wish to investigate

. 1P + 2P + ... + nP

bm

n->oo nP+l

here p is a positive integer. Letting Sn = 1P + 2P + · · · + nP and

Pn = nP+^1 , we may find, instead,

nP

n->oo lim nP+I - (n - l)P+l

nP

lim

n->oo (p + l)nP -^1 / 2 (p + l)pnP-1 + · · · - (-l)P+l

1

p+l

Since the last limit exists, the proposed limit also exists and equals

1/(p + 1).