# Sequences and Series/Print version

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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

- .

**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:**