Rit (Vísindafélag Íslendinga) - 01.06.1946, Page 39

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n becomes infinitely large. There exists, therefore, a limit,
although in the infinite region, for how often we can
00
differentiate the infinite series anxn without reducing
7! = 0
it to zero.
On the other hand our expression: -
y
n = 0
anx
is defined so that
t d" Z/n / \
an =-------=q>(n)
n! dx
(73)
(74)
where n is zero or a positive integer, and there is no upper
limit for n. Subsútuting from (70) in (73) we get
-B + Iy~a xr
S - Un + 1 + r A
r = 0
(75)
y = f (n,x) + x
or with regard to (71)
co —>
II = anxn + lim x'1+12Za„ + i-(-r x ^
/i = 0 n —>oo /* — 0
Frorn (76) in connexion with (73) we infer that
—> 00
(77)
/1 = 0 /i = 0
excepting, if
llm U" + ,X<I« + ,+,Vl-0 <77a)
n —> oo
/• =0