Page 286 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 11] INFINITE SERIES 277

@ @ 1 @ @
2
f ðx 0 þ h; y 0 þ kÞ¼ f ðx 0 ; y 0 Þþ h þ k f ðx 0 ; y 0 Þþ h þ k f ðx 0 ; y 0 Þþ
@x @y 2! @x @y
1 @ @ n ð18Þ
h þ k
n! @x @y
þ f ðx 0 ; y 0 Þþ R n
where
1 @ @ nþ1
h þ k f ðx 0 þ h; y 0 þ kÞ; 0 < < 1
ðn þ 1Þ! @x @y
R n ¼
and where the meaning of the operator notation is as follows:

@ @

h þ k f ¼ hf x þ kf y ;
@x @y
@ @ 2 2
2
h þ k ¼ h f xx þ 2hkf xy þ k f yy
@x @y
@ @
n
and we formally expand h þ k by the binomial theorem.
@x @y
Note:In alternate notation h ¼ x ¼ x x 0 , k ¼ y ¼ y y 0 .
If R n ! 0as n !1 then an unending continuation of terms produces the Taylor series for f ðx; yÞ.
Multivariable Taylor series have a similar pattern.

4. Double Series. Consider the array of numbers (or functions)
0 1
u 11 u 12 u 13 ...
u 21 u 22 u 23 ... C
B
B C
B ... C
u 31 u 32 u 33
. . . . . . . . .
@ A
m n
X X
u pq be the sum of the numbers in the first m rows and first n columns of this
Let S mn ¼
p¼1 q¼1
array. If there exists a number S such that lim S mn ¼ S,we say that the doubles series
m!1
n!1
1 1
X X
u pq converges to the sum S; otherwise, it diverges.
p¼1 q¼1
Definitions and theorems for double series are very similar to those for series already
considered.
n
Y
5. Infinite Products. Let P n ¼ð1 þ u 1 Þð1 þ u 2 Þð1 þ u 3 Þ ... ð1 þ u n Þ denoted by ð1 þ u k Þ, where
k¼1
we suppose that u k 6¼ 1; k ¼ 1; 2; 3; ... .If there exists a number P 6¼ 0 such that lim P n ¼ P,
n!1
1
Y
ð1 þ u k Þ,orbriefly
we say that the the infinite product ðð1 þ u 1 Þð1 þ u 2 Þð1 þ u 3 Þ ... ¼
ð1 þ u k Þ, converges to P; otherwise, it diverges. k¼1
If ð1 þju k jÞ converges, we call the infinite product ð1 þ u k Þ absolutely convergent.It can
be shown that an absolutely convergent infinite product converges and that factors can in such
cases be rearranged without affecting the result.
Theorems about infinite products can (by taking logarithms) often be made to depend on
theorems for infinite series. Thus, for example, we have the following theorem.
Theorem. A necessary and sufficient condition that ð1 þ u k Þ converge absolutely is that u k converge
absolutely.