Mathematical Physics Vol 1

5.1 Functional series. Power series

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3 Let the terms of a convergent series in the interval [ a , b ] have continuous derivatives (in the same interval). If the series of derivatives is uniformly convergent for x ∈ [ a , b ] , then the initial series will also be uniformly convergent in the same interval, and can be differentiated term-by-term, namely

∞ ∑ k = 0

∞ ∑ k = 0

d d x

f ′ k , for ∀ x ∈ [ a , b ] .

f ′ =

f k =

(5.7)

Power series

Definition A series of the form ∞ ∑ m = 0

m = a

2 + ··· ,

a m ( x − x 0 )

0 + a 1 ( x − x 0 )+ a 2 ( x − x 0 )

(5.8)

is called a power series . The constants a 0 , a 1 , ... , are called series coefficients .

It is assumed that the constants and the variable x belong to the set of real numbers (unless otherwise noted).

R Note that the power series is a functional series where f k ( x )= a k ( x − x 0 ) k . In the special case, when x 0 = 0, the power series has the following form ∞ ∑ m = 0 a m x m = a 0 + a 1 x + a 2 x 2 + ···

(5.9)

Theorem 11 (Abel’s theorem) If the series (5.9) is convergent for x = a , it is absolutely convergent for each x when | x | < | a | .

Definition For each power series (5.9) there exist a non-negative number R (including + ∞ ), such that the series is absolutely convergent for ∀ x ∈ ( − R , R ) , that is, for | x | < R , and divergent for ∀ x outside this interval. The number R is called the convergence radius , and the interval ( − R , R ) the convergence interval .

Operations on power series

1 ◦ Each power series, which is convergent for x ∈ ( − R , R ) , can be integrated in the interval [ 0 , x ] , where | x | < R , and in that case the integral of the sum is the sum of the integrals, namely x Z 0 f ( x ) d x = x Z 0 ∞ ∑ k = 0 a k x k ! d x = ∞ ∑ k = 0   x Z 0 a k x k d x   , for | x | < R . (5.10)