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

.


Clipboard

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: