University Mathematics Handbook (2015)

VI. Series

Chapter 4. Series of Functions

4.1  Sequences of Functions

a.   is a sequence of functions, and is their domain.

b.  The sequence of functions converges at if the sequence converges to . Written:

c.  Set of all points where the sequence of functions converges, is called domain of convergence of the sequence .

d.  For all of set , there is a target value . All limit values define the function , called limit function.

e.  The sequence of functions uniformly converges to function in , if for every , there exists (dependent of only), such that for all and for all , there holds .

f.  Criteria of Uniform Convergence

Cauchy's Criterion: sequence uniformly converges in domain if and only if there exists (dependent of only), such that for all and for all , and for all integer , there holds .

g.  Sequence of functions uniformly converges to in domain if and only if .

h.  If a sequence of continuous functions in uniformly converges in to , then is continuous in .

4.2  Series of Function

a.  Series , when is a sequence of factions defined in common domain is a series of functions.

b.  At a constant , the series is a series of numbers .

c.  The set of all points on where the series converges is called the domain of convergence of the series.

d.  The sum of the series, is a function defined in the series domain of convergence, and .

4.3  Uniform Convergence of Series

a.  The series is uniformly convergent in if the sequence of its partial sums is uniformly convergent in .

b.  Cauchy's Criterion: a series is uniformly convergent in if and only if there exists (dependent of only), such that for every for all , for all integer and for all , there holds .

c.  Weierstrass Test: if for series of functions defined in there exists a positive convergent series such that, starting from some , for all , then the series is uniformly convergent in .

d.  Given series .

1.  Dirichlet Test: if all partial sums of the series have common bound, that is, there exists such that for all and all , and the sequence is monotonic and uniformly convergent to zero at , then the given series is uniformly convergent in .

2.  Abel's Test: if the sequence is monotonic and bounded in , and the series is uniformly convergent in , then the given series, also, is uniformly convergent in .

4.4  Continuity, Derivability and Integrability of Sums of Series

a.  If functions are continuous in domain and the series is uniformly convergent in , then function is continuous in .

b.  If the sum of a series of continuous functions converges to a discontinuous function in the same domain, then the convergence is not uniform.

c.  If is a series of functions continuous in , and the series of functions uniformly converges to on , then:

d.  If functions are derivable and have continuous derivatives on interval , the series is convergent on and series of derivatives is uniformly convergent on , then:

In other words, the derivative of a sum equals the sum of derivatives.

4.5  Power Series and Radius of Convergence

A functional series in the form of , or is called a power series.

Substituting in the latter series, will result in the former series.

a.  Radius of Convergence Existence Theorem

For all power series there exists non-negative , , such that for all which holds , the series is convergent, and for all which holds , the series is divergent, and if , the series converges at only. If , the series converges for all . is called the radius of convergence of the series.

b.  Formulas for Calculating the Radius of Convergence

1.  

2.  Cauchy-Hadamard formula:

4.6  Uniform Convergence, Derivation and Integration of Power Series

a.  If power series with a convergence radius :

1.  Diverges at endpoint , then the convergence on interval is not uniform.

2.  Converges at endpoint , , then the convergence is uniform on intervals , .

b.  The series is uniformly convergent on every segment , which is entirely within .

c.  Sum of power series with radius of convergence is a function continuous on all . If, in addition, it converges on , , then is continuous from the left (from the right) at endpoint , .

d.  Let be the radius of convergence of a power series . Then, for all , there holds:

1.  .

2.  Both series have the same radius of convergence .

3.  If a power series converges at , , then, their integrals series also converges at , .

e.  If Let be the radius of convergence of a power series. Then, for all , there holds:

1.  

2.  The power series and derivatives series have the same power of convergence .

3.  If the series of derivatives converges at , , then, the original series converges at the same endpoint.

4.7  Power Series Expansion of Functions

Function defined on is expanded to a power Taylor series, if there exists a power series converging to on .

a.  A Necessary Condition of Expansion to Power Series: if is the sum of series on , then is infinitely derivable and all its derivatives are functions contiguous on .

b.  The Expansion of to Power (Binomial) Series in Powers of :

converging at .

c.  Examples:

1.  

2.  

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6.  

7.