matan27-30 by Erbolat

27. Properties of Function Defined by Power Series.

We know the properties of functions defined by Power series. Henceforth, we consider only power series with non-zero radius of convergence.

Theorem 3. A power series (1) with R>0 is continuous and differentiable in its interval of convergence, and it's derivative can be obtained by differentiating, term by term; that is (2) which can also be written as

(3)

This series also has radius of convergence R.

Proof: First, the series in (2) and (3) are came, since the after is obtained by shifting the index of summation in the former. Since

the radius of convergence of the power series in (3) is R. Therefore, the power series, in (3) convergence uniformly in every interval [x₀-r, x₀+r] such that 0<r<R and (3) (x₀-R, x₀+R).

Theorem 4. A power series R>0, has derivative of all orders in it's interval of convergence, which can be obtained by repeated term by term differentiation, thus

Corollary. If then

28. Uniqueness of Power Series

Theorem 5. (1) for all x in so me interval (x₀-r, x₀+r), then a_n=b_n (2)

Proof: Let and

From corollary ( if then )

and (3)

From (1), in (x₀-r, x₀+r). Therefore, n≥0. This and (3) imply (2).

Theorem 6. If x1 and x2 are in the interval of convergence of (4) then (5) that is a power series may be integrated term by term between any the points in it's interval convergence.

29.Taylor's series.

A function must have derivatives of all orders in some neighborhood(окрестность) of x₀ and the only power series in (x-x₀) can be possibly coverge to such neighborhood is (1)

This is called the Taylor's series of about (also Maclouzin series of if )

The n-th partial sum of (6) is the Taylor polynomial

If is infinitely differentiable on (a,b) and x and x₀ are in (a,b), then for every integer n≥0 there c_n between x and x₀

for all x in (a,b), is iff

Thorem. Suppose that infinitely differentiable on an interval G and

(2)

Then, if the Taylor series uniform convergence in

Thorem. If (3)

(4) and α and β constants, then

, where

Thorem. If and are given by (3) and (4) then

(5) where

and

30. Abel's theorem.

We know that a function defined by a convergent power series

(1)

is continuous in the open interval ()

The next theorem concerns the behavior(поведение) of as x approaches an endpoint of the interval of convergence.

Theorem Abel's.

Let be defined by a power series (1) with finite radius of convergence R

a) If , then

Proof: We consider a simpler problem first. Let and (finite) we will show that (2)

(3), where S_n=b₀+b₁+...+b_n

Since and therefore (4) we can multiply through by S and write . Subtracting this from (3) yields .

if choose N so that , if n>N+1.

Then, if 0<y<1

because of the second equality in (4).

Therefore, , if this proves (2).

to obtain (a) from this, let and ;

to obtain (b) and