Sequences and Series/Print version


Sequences and Series

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Multiple limits

Theorem (interchanging summation and integration):

Let   be a measure space, and let   be a sequence of functions from   to  , where   or  . If either of the two expressions

  or  

converges, so does the other, and we have

 .

Proof: Regarding the summation as integration over   with σ-algebra   and counting measure, this theorem is an immediate consequence of Fubini's theorem, given that integration and summation are defined pointwise.  

Theorem (interchanging summation and real differentiation):

Let   be a sequence of continuously differentiable functions from an open subset   of   to  . Suppose that both

  and  

converge for all  , and that for all   there exists   and a sequence   in   such that

  and  .

Then

 

for all  .

Proof:  


Series and integration

Theorem (Abelian partial summation):

Let   be a sequence of complex numbers, and let   be differentiable on  . Finally define

 .

Then for   we have

 .

Proof: If  , we have

 

But

 

so that

 .  


Power series

Proposition (identity theorem for one-dimensional power series):

Let

  and  

be two (complex or real) power series that converge on   for some  . Suppose that   is an accumulation point of the set  . Then we have   for all  .

Proof: Assume that not   for all  . Then there exists a least   (call it  ) such that  . Consider the function

 ,

which is defined on at least  . Since   for  , the power series   starts at  . Therefore,

 

is a well-defined function on   which is also continuous due to the continuity of power series. Moreover,

 ,

and by continuity of  , there exists a   such that   for all  . But by definition,

 ,

so that we have for   that   and consequently  , and hence  . But this contradicts the assumption that   was an accumulation point of  .  

Example (falsity of the identity theorem for multi-dimensional power series):

For multi-dimensional power series, that is power series of the type

  for a  ,

the set   may have   as an accumulation point even when   does not vanish. An easy example (which works in any dimension  ) is   and

 .


 

To do:
The LHS needs to converge to   as   is chosen in the right way.


Theorem (Abel's theorem):

Let

 

be a real or complex power series of convergence radius  , and suppose that

 .

Then

 .

{{proof|By Abelian partial summation, we have

 

for   and  , where we denote as usual

 .

Substituting  , we get

 .

We then put


Dirichlet‒Hurwitz series

Definition (Dirichlet‒Hurwitz series):

Let   be a function, and let  . The Dirichlet‒Hurwitz series associated to   and   is the function of   given by the series

 .

Definition (abscissa of absolute convergence of Dirichlet‒Hurwitz series):

Let   be a function, and let  . Suppose that there exists a number   such that

 

converges whenever   and diverges whenever  . Then   is called the abscissa of absolute convergence of the Dirichlet‒Hurwitz series associated to   and  .

Proposition (existence of abscissa of absolute convergence of Dirichlet‒Hurwitz series):

Let   be a function, and let  . Suppose that


Infinite products

Definition (infinite product):

Let   be a sequence of numbers in   or  . If the limit

 

exists, it is called the infinite product of   and denoted by

 .

Proposition (necessary condition for convergence of infinite products):

In order for the infinite product

 

of a sequence   to exist and not to be zero, it is necessary that

 .

Proof: Suppose that not  . Then there exists   and an infinite sequence   such that for all   we have  . Thus, upon denoting

 ,

we will have

 .

Suppose for a contradiction that   exited and was equal to  . Then when   is sufficiently large, we will have

 ,

which is a contradiction.  

Proposition (series criterion for the convergence of infinite products):

Let   be a sequence of real numbers. If

 ,

then

 

converges.

Proof: