Rit (Vísindafélag Íslendinga) - 01.06.1951, Page 5
5
(T
'v's"
1 a
X X2 X'
froin which we are inclined to infer that thc series becomes
infinitely large for x=o. We have to be on guard against
such inferences. If the series is divergent the sutn of the
first terms indicates by no means the real value of the in-
finite series. On the contrary, the sum of the first terms is
frequently digressing from it or divergently oscillating about it.
Obviously we liave the following invariant transformations
r
a"+T
(•v-y)"
(a — y)n+'
(?—y)"
(x—y)',+/
(5)
n 0 n- 0
where y is an arbitrary quantity.
The fransformations shown by (5) may happen to be of
some value. Specific examples will, however, not be given
here, as I wish to treat the whole matter in a more general
way by means of two symbols for mathematical operations.
The symbols in question are very well known, my notations
are, however, somewhat at variance with the common us-
age of the symbols. I shall, therefore, briefly explain my
mode of signification.
The two symbols are:—
1. The differential symbol Ðx
in that
Ðx->f(x)-
further
df(x)
dx
dx
= Ðxf(x)
Ð:->f(x)=dn£*)=Ðnxf(x)
and
2. The difference symbol A.>r
in that
a x f(x)=f(x+l)—f(x)= a .v f(x)
further
■ n!( - í)n~rf(x \ -r)
r!(n—r)!
(6)
A "f(x)= A ” —^ f(x)—
(7)