Calculus 3 3: chuỗi hội tụ

Then, P anxn converges uniformly in any closed interval which lies entirely within its convergence, and lim... Then, P anxn converges uniformly in any closed interval which lies entirely

Infinite Series and Differential Equations

Nguyen Thieu Huy

Hanoi University of Science and Technology

Calculation of radius of convergence

The note can be applied to the series of the form P an(f (x ))n by putting

X = f (x ) and reducing it to the power series P anXn

Calculation of radius of convergence

The note can be applied to the series of the form P an(f (x ))n by putting

X = f (x ) and reducing it to the power series P anXn

Calculation of radius of convergence

2 To calculate the domain of conv.for P anxn we can either compute

as previously, or compute the radius of conv R, and then interval of

conv (−R, R) Then, check the two endpoints −R and R to decide

whether they can be included in the domain of conv Outside the

interval of conv (i.e., for |x | > R) we knew that the series is div

The note can be applied to the series of the form P an(f (x ))n by putting

X = f (x ) and reducing it to the power series P anXn

Calculation of radius of convergence

n→∞

n

p|an|) we have that theradius of convergence is R = 1ρ with the conventions 10 = ∞ and ∞1 = 0.Note:

1 P anxnis absolutely convergent ∀x ∈ (−R, R)

2 To calculate the domain of conv.for P anxn we can either compute

as previously, or compute the radius of conv R, and then interval ofconv (−R, R) Then, check the two endpoints −R and R to decidewhether they can be included in the domain of conv Outside theinterval of conv (i.e., for |x | > R) we knew that the series is div.The note can be applied to the series of the form P an(f (x ))n by putting

Properties of power series

Uniform convergence

Let power series P anxn have radius of convergence R > 0 Then, P anxn

converges uniformly in any closed interval which lies entirely within its

convergence, and lim

Properties of power series

Uniform convergence

Let power series P anxn have radius of convergence R > 0 Then, P anxn

converges uniformly in any closed interval which lies entirely within its

interval of convergence (−R, R)

Proof Fix any [a, b] ⊂ (−R, R), then −R < a < b < R So, ∃x0 ∈ (0, R)

such that [a, b] ⊂ [−x0, x0] We have power series converges absolutely at

convergence, and lim

Properties of power series

Uniform convergence

Let power series P anxn have radius of convergence R > 0 Then, P anxn

converges uniformly in any closed interval which lies entirely within itsinterval of convergence (−R, R)

Proof Fix any [a, b] ⊂ (−R, R), then −R < a < b < R So, ∃x0 ∈ (0, R)such that [a, b] ⊂ [−x0, x0] We have power series converges absolutely at

x0 Moreover, |anxn| 6 |anxn

0| ∀x ∈ [a, b], ∀n, andP |anxn

0| is convergent.Thanks to Weierstrass, power series converges uniformly on [a, b]

Corollary 2 (Integrability)

Power series is integrable on any closed interval lying entirely within its

interval of convergence, and

Remark When a power series converges up to and including an endpoint

of its interval of convergence, the interval of uniform convergence alsoextends so far as to include this endpoint Therefore the above propertiescan be extended to include that endpoint

Corollary 2 (Integrability)

Power series is integrable on any closed interval lying entirely within its

interval of convergence, and

Remark When a power series converges up to and including an endpoint

of its interval of convergence, the interval of uniform convergence alsoextends so far as to include this endpoint Therefore the above propertiescan be extended to include that endpoint

Corollary 2 (Integrability)

Power series is integrable on any closed interval lying entirely within its

interval of convergence, and

Proof Based on the fact that the series of derivatives P nanxn−1 has the

same radius of convergence as the original series P anxn

Remark When a power series converges up to and including an endpoint

of its interval of convergence, the interval of uniform convergence alsoextends so far as to include this endpoint Therefore the above propertiescan be extended to include that endpoint

Remark When a power series converges up to and including an endpoint

of its interval of convergence, the interval of uniform convergence alsoextends so far as to include this endpoint Therefore the above propertiescan be extended to include that endpoint

Expansion of Functions in Power Series

Problem: Let f be infinitely many times differentiable on

2 Suppose such a power series as above exists, is it unique?

Solution: (2) Suppose there exists a power series

∞

P

n=0

an(x − x0)n suchthat f (x ) =

Expansion of Functions in Power Series

Problem: Let f be infinitely many times differentiable on

2 Suppose such a power series as above exists, is it unique?

Solution: (2) Suppose there exists a power series

∞

P

n=0

an(x − x0)n suchthat f (x ) =

Expansion of Functions in Power Series

Problem: Let f be infinitely many times differentiable on

2 Suppose such a power series as above exists, is it unique?

Solution: (2) Suppose there exists a power series

∞

P

n=0

an(x − x0)n suchthat f (x ) =

Expansion of Functions in Power Series

Problem: Let f be infinitely many times differentiable on

2 Suppose such a power series as above exists, is it unique?

Solution: (2) Suppose there exists a power series

∞

P

n=0

an(x − x0)n suchthat f (x ) =

Expansion of Functions in Power series

∞Pn=0

f (n) (x 0 ) n! (x − x0)n is called Taylor’s Seriesof f

For x0= 0, the series

∞

P

n=0

f (n) (0) n! xn is called Maclaurin’s Series of f

Expansion of Functions in Power series

Assume f is infinitely many times differentiable on (x0− R, x0+ R) Then,

the power series

∞Pn=0

f (n) (x 0 ) n! (x − x0)n is called Taylor’s Seriesof f

For x0= 0, the series

∞

P

n=0

f (n) (0) n! xn is called Maclaurin’s Series of f

Expansion of Functions in Power series

f (n) (x 0 ) n! (x − x0)n is called Taylor’s Seriesof f For x0= 0, the series

∞

P

n=0

f (n) (0) n! xn is called Maclaurin’s Series of f

Expansion of Functions in Taylor’s Series, Maclaurin’s Series

Theorem

Let f be infinitely many times differentiable (x0− R, x0+ R) and

satisfying ∃M > 0 such that |f(n)(x )| 6 M ∀x ∈ (x0− R, x0+ R) ∀n ∈ N

Then, Taylor’s series of f converges to f on (x0− R, x0+ R), this means

So, |f (x ) − Sk(x )| = |f(k+1)!(k+1)(ξ)(x − x0)k+1| 6 MRk+1

Therefore, lim

k→∞Sk(x ) = f (x ) ∀x ∈ (x0− R, x0+ R) (q.e.d)

Remark: The above theorem still holds true in case R = ∞ In the proof

we just take arbitrary finite subinterval to have necessary estimates

Expansion of Functions in Taylor’s Series, Maclaurin’s Series

Theorem

Let f be infinitely many times differentiable (x0− R, x0+ R) and

satisfying ∃M > 0 such that |f(n)(x )| 6 M ∀x ∈ (x0− R, x0+ R) ∀n ∈ N

Then, Taylor’s series of f converges to f on (x0− R, x0+ R), this means

Remark: The above theorem still holds true in case R = ∞ In the proof

we just take arbitrary finite subinterval to have necessary estimates

Expansion of Functions in Taylor’s Series, Maclaurin’s Series

Theorem

Let f be infinitely many times differentiable (x0− R, x0+ R) and

satisfying ∃M > 0 such that |f(n)(x )| 6 M ∀x ∈ (x0− R, x0+ R) ∀n ∈ N.Then, Taylor’s series of f converges to f on (x0− R, x0+ R), this means

Remark: The above theorem still holds true in case R = ∞ In the proof

we just take arbitrary finite subinterval to have necessary estimates

Example 1: Expand to Maclaurin’s Series for f (x ) = sin x ∀x ∈ R

f0(x ) = cos x ; f00(x ) = − sin x ; f000(x ) = − cos x ; f(4)(x ) = sin x ;

We have f(n)(x ) = (sin x )(n)= sin(x + nπ2 ) ∀x ∈ R

(−1) k x 2k+1

(2k+1)!

Example 2: Similarly, cos x =

∞Pk=0

(−1)kx2k(2k)!

Example 3: Easy to seeex =

∞Pk=0

=cos ϕ + i sin ϕ

Example 1: Expand to Maclaurin’s Series for f (x ) = sin x ∀x ∈ R

f0(x ) = cos x ; f00(x ) = − sin x ; f000(x ) = − cos x ; f(4)(x ) = sin x ;

We have f(n)(x ) = (sin x )(n)= sin(x + nπ2 ) ∀x ∈ R

(−1) k x 2k+1

(2k+1)!

Example 2: Similarly, cos x =

∞Pk=0

(−1)kx2k(2k)!

Example 3: Easy to see ex =

∞Pk=0

=cos ϕ + i sin ϕ

Example 1: Expand to Maclaurin’s Series for f (x ) = sin x ∀x ∈ R

f0(x ) = cos x ; f00(x ) = − sin x ; f000(x ) = − cos x ; f(4)(x ) = sin x ;

We have f(n)(x ) = (sin x )(n)= sin(x + nπ2 ) ∀x ∈ R

(−1) k x 2k+1

(2k+1)!

Example 2: Similarly, cos x =

∞Pk=0

(−1) k x 2k

(2k)!

Example 3: Easy to see ex =

∞Pk=0

=cos ϕ + i sin ϕ

Example 1: Expand to Maclaurin’s Series for f (x ) = sin x ∀x ∈ R

f0(x ) = cos x ; f00(x ) = − sin x ; f000(x ) = − cos x ; f(4)(x ) = sin x ;

We have f(n)(x ) = (sin x )(n)= sin(x + nπ2 ) ∀x ∈ R

(−1) k x 2k+1

(2k+1)!

Example 2: Similarly, cos x =

∞Pk=0

(−1) k x 2k

(2k)!

Example 3: Easy to see ex =

∞Pk=0

Example 1: Expand to Maclaurin’s Series for f (x ) = sin x ∀x ∈ R

f0(x ) = cos x ; f00(x ) = − sin x ; f000(x ) = − cos x ; f(4)(x ) = sin x ;

We have f(n)(x ) = (sin x )(n)= sin(x + nπ2 ) ∀x ∈ R

(−1) k x 2k+1

(2k+1)!

Example 2: Similarly, cos x =

∞Pk=0

(−1) k x 2k

(2k)!

Example 3: Easy to see ex =

∞Pk=0

Example 1: Expand to Maclaurin’s Series for f (x ) = sin x ∀x ∈ R

f0(x ) = cos x ; f00(x ) = − sin x ; f000(x ) = − cos x ; f(4)(x ) = sin x ;

We have f(n)(x ) = (sin x )(n)= sin(x + nπ2 ) ∀x ∈ R

(−1) k x 2k+1

(2k+1)!

Example 2: Similarly, cos x =

∞Pk=0

(−1) k x 2k

(2k)!

Example 3: Easy to see ex =

∞Pk=0

x k

k!.Application: Euler’s formula: substituting x = i ϕ we obtain

SOME IMPORTANT POWER SERIES

Fourier Series

Periodic functions:

A function f (x ) is said to beperiodic with period T if for all x ∈ R,

f (x + T ) = f (x ), where T is a positive constant The least value of T > 0

is called the least period or simply the period of f (x )

Examples: Functions sin x , cos x are periodic with period T = 2π,

whereas, cosπx

L and sinπx

L are periodic with period 2L

A periodic function with period T is called a T -periodic function

Fourier Series

Periodic functions:

A function f (x ) is said to beperiodic with period T if for all x ∈ R,

f (x + T ) = f (x ), where T is a positive constant The least value of T > 0

is called the least period or simply the period of f (x )

Examples: Functions sin x , cos x are periodic with period T = 2π,

whereas, cosπx

L and sinπx

L are periodic with period 2L

A periodic function with period T is called a T -periodic function

Piecewise continuous functions

Definition

A function f (x ) is said to bepiecewise continuous on (a, b) if (a, b) can besubdivided into a finite number of subinterval a = x0< x1 < · · · < xn= bsuch that on each open subinterval (xj, xj +1) the function is continuous

∀j = 0, n − 1 and at each endpoint xj the limits from the right and fromthe left exist

We denote limit from the right lim

x →x j ; x >x j

f (x ) := f (xj + 0)and limit from the left lim

x →xj; x <x j

f (x ) := f (xj − 0)

2 Suppose such a series as above exists, is it unique?

Solution: (2): Suppose that there exists a series as above such that

2 Suppose such a series as above exists, is it unique?

Solution: (2): Suppose that there exists a series as above such that

2 Suppose such a series as above exists, is it unique?

Solution: (2): Suppose that there exists a series as above such that

am = 1L

LR

−L

f (x ) sinmπxL dx for all m = 1, 2, · · ·

Therefore, if such a series exists, then it is unique (because all an, bn areuniquely determined by f )

−L

f (x )dx.For m > 0: using formulas

am = 1L

LR

−L

f (x ) sinmπxL dx for all m = 1, 2, · · ·

Therefore, if such a series exists, then it is unique (because all an, bnareuniquely determined by f )

−L

f (x )dx.For m > 0: using formulas

−L

f (x ) sinmπxL dx for all m = 1, 2, · · ·

Therefore, if such a series exists, then it is unique (because all an, bn areuniquely determined by f )

−L

f (x )dx.For m > 0: using formulas

f (x ) sinmπxL dx for all m = 1, 2, · · ·

Therefore, if such a series exists, then it is unique (because all an, bn areuniquely determined by f )

−L

f (x )dx.For m > 0: using formulas

f (x ) sinmπxL dx for all m = 1, 2, · · ·

Therefore, if such a series exists, then it is unique (because all an, bn areuniquely determined by f )

Let f (x ) be defined on R, periodic with period 2L, and piecewise

continuous on (−L, L) Then, the series of trigonometric functions

f (x ) sinnπxL dx for all n = 1, 2, · · ·

is called the Fourier Seriesof f

The above-defined numbers an, bn are calledFourier Coefficients of f

Let f (x ) be defined on R, periodic with period 2L, and piecewise

continuous on (−L, L) Then, the series of trigonometric functions

f (x ) sinnπxL dx for all n = 1, 2, · · ·

is called the Fourier Seriesof f

The above-defined numbers an, bn are calledFourier Coefficients of f