Mathematical Physics Vol 1

Chapter 5. Series Solutions of Differential Equations. Special functions

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If this limit value does not exists, then we say that the series is divergent .

If the series (5.1) is convergent for all values of the variable x ∈ [ a , b ] , then the sum of the series represents a function f ( x ) , for x ∈ [ a , b ] , and can be represented in the following form f ( x )= S n ( x )+ R n ( x ) , (5.4) where S n is the partial sum, and R n ( x ) the remainder. Then f ( x )= lim n → ∞ S n ( x ) , lim n → ∞ R n ( x )= lim n → ∞ [ f ( x ) − S n ( x )]= 0 , (5.5) or | f − S n | = | R n ( x ) | < ε for each n ≥ N ( ε , x ) and for ∀ x ∈ [ a , b ] .

Definition

The series

∞ ∑ k = 0

f k ( x )

is absolutely convergent for some x = x 1 ∈ [ a , b ] , if the series ∞ ∑ k = 0 | f k ( x 1 ) | is convergent.

Definition The series (5.1) is uniformly convergent in the interval [ a , b ] , if for each arbitrary small ε > 0 there exists a positive integer N ( ε ) , which does not depend on x , such that | R n ( x ) | < ε for ∀ n ≥ N ( ε ) and ∀ x ∈ [ a , b ] .

R Note that a convergent series need not be uniformly convergent in the same interval.

Properties of uniformly convergent series

1 If the terms of a uniformly convergent (infinite) series are continuous functions of an independent variable x ∈ [ a , b ] , then their sum is also a continuous function of the variable x , in the same interval. 2 If the terms of a uniformly convergent series are continuous functions of an independent variable x ∈ [ a , b ] , then the integral of their sum is equal to the sum of their integrals, namely b Z a f d x = b Z a " ∞ ∑ k = 0 f k # d x = ∞ ∑ k = 0 b Z a f k d x , for ∀ x ∈ [ a , b ] . (5.6)