Page 282 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 11] INFINITE SERIES 273

1 1 1
X X X
n
n
Theorem 14. Two power series, for example, a n x and b n x , can be multiplied to obtain c n x n
n¼0 n¼0 n¼0
where
c n ¼ a 0 b n þ a 1 b n 1 þ a 2 b n 2 þ þ a n b 0 ð11Þ
the result being valid for each x within the common interval of convergence.
1
X n n
Theorem 15. If the power series a n x is divided by the power series b n x where b 0 6¼ 0, the quotient
n¼0
can be written as a power series which converges for sufficiently small values of x.
1 1
X X
n
n
Theorem 16. If y ¼ a n x , then by substituting x ¼ b n y ,we can obtain the coefficients b n in
n¼0 n¼0
terms of a n . This process is often called reversion of series.

EXPANSION OF FUNCTIONS IN POWER SERIES
This section gets at the heart of the use of infinite series in analysis. Functions are represented
through them. Certain forms bear the names of mathematicians of the eighteenth and early nineteenth
century who did so much to develop these ideas.
A simple way (and one often used to gain information in mathematics) to explore series representa-
tion of functions is to assume such a representation exists and then discover the details. Of course,
whatever is found must be confirmed in a rigorous manner. Therefore, assume
2 n
f ðxÞ¼ A 0 þ A 1 ðx cÞþ A 2 ðx cÞ þ þ A n ðx cÞ þ
Notice that the coefficients A n can be identified with derivatives of f .In particular
1 1
0 00 f ðnÞ ðcÞ; ...
2! n!
A 0 ¼ f ðcÞ; A 1 ¼ f ðcÞ; A 2 ¼ f ðcÞ; ... ; A n ¼
This suggests that a series representation of f is
1 1
0 00 2 f ðnÞ n
2! n!
f ðxÞ¼ f ðcÞþ f ðcÞðx cÞþ f ðcÞðx cÞ þ þ ðcÞðx cÞ þ
A first step in formalizing series representation of a function, f , for which the first n derivatives exist,
is accomplished by introducing Taylor polynomials of the function.
P 1 ðxÞ¼ f ðcÞþ f ðcÞðx cÞ;
0
P 0 ðxÞ¼ f ðcÞ
1
2
00
0 f ðcÞðx cÞ ;
2!
P 2 ðxÞ¼ f ðcÞþ f ðcÞðx cÞþ
1
f
0 ðnÞ n
n!
P n ðxÞ¼ f ðcÞþ f ðcÞðx cÞþ þ ðcÞðx cÞ ð12Þ
TAYLOR’S THEOREM
Let f and its derivatives f ; f ; ... ; f ðnÞ exist and be continuous in a closed interval a x b and
00
0
suppose that f ðnþ1Þ exists in the open interval a < x < b. Then for c in ½a; bŠ,
f ðxÞ¼ P n ðxÞþ R n ðxÞ;
where the remainder R n ðxÞ may be represented in any of the three following ways.
For each n there exists such that