Infinite Series and Differential Equations

Nguyen Thieu Huy

Hanoi University of Science and Technology

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

1 / 15

Series of Functions

Definition

Let {un (x)}∞

n=1 be a sequence of functions defined ∀x ∈ D ⊂ R. Formal

sum

∞

un (x) is called a series of functions.

n=1

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

2 / 15

Series of Functions

Definition

Let {un (x)}∞

n=1 be a sequence of functions defined ∀x ∈ D ⊂ R. Formal

sum

∞

un (x) is called a series of functions.

n=1

∞

Here, when x is taken a concrete real value x0 ∈ D then

un (x0 ) is a

n=1

series of numbers.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

2 / 15

Series of Functions

Definition

Let {un (x)}∞

n=1 be a sequence of functions defined ∀x ∈ D ⊂ R. Formal

sum

∞

un (x) is called a series of functions.

n=1

∞

Here, when x is taken a concrete real value x0 ∈ D then

un (x0 ) is a

n=1

series of numbers.

∞ n

2 (x+1)n

Ex. 1:

, ∀x ∈ R, is a series of functions. Substituting x by 3,

n!

n=1

∞

we obtain

n=1

8n

n!

is a series of numbers.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

2 / 15

Series of Functions

Definition

Let {un (x)}∞

n=1 be a sequence of functions defined ∀x ∈ D ⊂ R. Formal

sum

∞

un (x) is called a series of functions.

n=1

∞

Here, when x is taken a concrete real value x0 ∈ D then

un (x0 ) is a

n=1

series of numbers.

∞ n

2 (x+1)n

Ex. 1:

, ∀x ∈ R, is a series of functions. Substituting x by 3,

n!

n=1

∞

we obtain

n=1

8n

n!

is a series of numbers.

∞

Ex. 2: Series of functions

∞

n=1

n=1

√1

n

1

nx ,

∀x ∈ R. Substituting x by 12 , we have

is a series of numbers.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

2 / 15

Domain of Convergence

Definition

∞

un (x), the set X ⊂ R is called

∞

un (x) is convergent ∀x ∈ X

n=1

domain of convergence ⇐⇒ ∞

un (x) is divergent ∀x ∈

/ X.

For series of functions

n=1

n=1

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

3 / 15

Domain of Convergence

Definition

∞

un (x), the set X ⊂ R is called

∞

un (x) is convergent ∀x ∈ X

n=1

domain of convergence ⇐⇒ ∞

un (x) is divergent ∀x ∈

/ X.

For series of functions

n=1

n=1

∞

In that case, for each x ∈ X we put S(x) :=

un (x) ∈ R, then we

n=1

obtain a function x → S(x). The function S(x) is called the sum of the

∞

series of functions

∞

un (x), and we say

n=1

un (x) is convergent on X to

n=1

the function S(x).

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

3 / 15

Domain of Convergence

Definition

∞

un (x), the set X ⊂ R is called

∞

un (x) is convergent ∀x ∈ X

n=1

domain of convergence ⇐⇒ ∞

un (x) is divergent ∀x ∈

/ X.

For series of functions

n=1

n=1

∞

In that case, for each x ∈ X we put S(x) :=

un (x) ∈ R, then we

n=1

obtain a function x → S(x). The function S(x) is called the sum of the

∞

series of functions

∞

un (x), and we say

n=1

un (x) is convergent on X to

n=1

the function S(x).

∞

Ex. 1: Domain of convergence for

1

1−x

∞

=

x n is (−1, 1). Its sum is

n=0

x n;

∀x ∈ (−1, 1).

n=0

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

3 / 15

∞

Ex. 2: Domain of convergence for

∞

Its sum S(x) :=

n=1

Nguyen Thieu Huy (HUST)

n=1

1

nx

1

nx

is (1, ∞) (property of p-Series).

is called ζ-Riemann function.

Infinite Series and Diff. Eq.

4 / 15

∞

Ex. 2: Domain of convergence for

∞

Its sum S(x) :=

n=1

n=1

1

nx

1

nx

is (1, ∞) (property of p-Series).

is called ζ-Riemann function.

♣ The previous tests can be applied to find domains of convergence.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

4 / 15

∞

Ex. 2: Domain of convergence for

∞

Its sum S(x) :=

n=1

n=1

1

nx

1

nx

is (1, ∞) (property of p-Series).

is called ζ-Riemann function.

♣ The previous tests can be applied to find domains of convergence.

∞

Ex. 3: Find the Domain of Convergence for

n=1

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

xn

n ;

un (x) =

xn

n .

4 / 15

∞

Ex. 2: Domain of convergence for

∞

Its sum S(x) :=

n=1

n=1

1

nx

1

nx

is (1, ∞) (property of p-Series).

is called ζ-Riemann function.

♣ The previous tests can be applied to find domains of convergence.

∞

Ex. 3: Find the Domain of Convergence for

n=1

We have lim

n→∞

un+1 (x)

un (x)

= lim

n→∞

x n+1

n+1

xn

n

xn

n ;

|x|n

n→∞ n+1

= lim

un (x) =

xn

n .

= |x|.

Therefore, according to Ratio Test

1

If |x| < 1 then series is conv.

2

If |x| > 1 then series is div.

3

Remain to consider |x| = 1 (here Ratio Test cannot be applied)

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

4 / 15

∞

Ex. 2: Domain of convergence for

∞

Its sum S(x) :=

n=1

n=1

1

nx

1

nx

is (1, ∞) (property of p-Series).

is called ζ-Riemann function.

♣ The previous tests can be applied to find domains of convergence.

∞

Ex. 3: Find the Domain of Convergence for

n=1

We have lim

n→∞

un+1 (x)

un (x)

= lim

n→∞

x n+1

n+1

xn

n

xn

n ;

|x|n

n→∞ n+1

= lim

un (x) =

xn

n .

= |x|.

Therefore, according to Ratio Test

1

If |x| < 1 then series is conv.

2

If |x| > 1 then series is div.

3

Remain to consider |x| = 1 (here Ratio Test cannot be applied)

∞ 1

n=1 n ⇒ div. (p-Series with p = 1)

(−1)n

becomes ∞

n=1 n . This is alternating series

1

1

n ∀n and 2) limn→∞ n = 0. So, it is conv.

a) For x = 1: Series becomes

b) For x = −1: Series

1

satisfying 1) n+1

Altogether, Domain of convergence is [−1; 1).

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

4 / 15

Convergence: revisited

∞

un (x) be convergent on D to function S(x). Then, using the

Let

n=1

language ( , N) we can rewrite the notion of convergence as follows:

∞

un (x) is convergent on D to function S(x)

Series

n=1

k

⇔ ∀x ∈ D; ∀ > 0, ∃Nx; ∈ N such that |S(x) −

un (x)| <

∀k > Nx; .

n=1

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

5 / 15

Convergence: revisited

∞

un (x) be convergent on D to function S(x). Then, using the

Let

n=1

language ( , N) we can rewrite the notion of convergence as follows:

∞

un (x) is convergent on D to function S(x)

Series

n=1

k

⇔ ∀x ∈ D; ∀ > 0, ∃Nx; ∈ N such that |S(x) −

un (x)| <

∀k > Nx; .

n=1

Ex. :

1

∞

Series

x n is convergent on (−1, 1) to function

n=0

∞

2

Series

n=1

∞

1

nx

S(x) =

n=1

1

1−x

convergent on (1, ∞) to ζ-Riemann function

1

nx .

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

5 / 15

Convergence: revisited

∞

un (x) be convergent on D to function S(x). Then, using the

Let

n=1

language ( , N) we can rewrite the notion of convergence as follows:

∞

un (x) is convergent on D to function S(x)

Series

n=1

k

⇔ ∀x ∈ D; ∀ > 0, ∃Nx; ∈ N such that |S(x) −

un (x)| <

∀k > Nx; .

n=1

Ex. :

1

∞

Series

x n is convergent on (−1, 1) to function

n=0

∞

2

Series

n=1

∞

1

nx

S(x) =

n=1

1

1−x

convergent on (1, ∞) to ζ-Riemann function

1

nx .

Convergence according to the above definition is sometimes called

pointwise convergence. To do convenient calculi on a series of functions,

we need the concept of uniform convergence.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

5 / 15

Uniform convergence

Definition

∞

un (x) is called uniformly convergent on D to function S(x)

Series

n=1

⇔ ∀x ∈ D; ∀ > 0, ∃N ∈ N depending only on , not on x, such that

k

|S(x) −

un (x)| <

∀k > N .

n=1

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

6 / 15

Uniform convergence

Definition

∞

un (x) is called uniformly convergent on D to function S(x)

Series

n=1

⇔ ∀x ∈ D; ∀ > 0, ∃N ∈ N depending only on , not on x, such that

k

|S(x) −

un (x)| <

∀k > N .

n=1

We introduce the following two tests to check the uniform convergence:

Cauchy test for uniformly convergent Series of functions

∞

un (x) and set D ⊂ R satisfying

Consider series

n=1

m

∀x ∈ D; ∀ > 0 ∃N ∈ N such that |

∞

series

un (x)| <

∀m, k > N . Then,

n=k+1

un (x) is uniformly convergent on D.

n=1

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

6 / 15

Weierstrass test

∞

un (x) and set D ⊂ R satisfying ∃ non-negative

Consider series of

n=1

numbers an ; n = 1, 2, · · ·

1

|un (x)|

an ∀x ∈ D; ∀n = 1, 2, · · · .

2

Series of numbers

∞

an is convergent.

n=1

∞

Then, Series

un (x) is uniformly convergent on D.

n=1

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

7 / 15

Weierstrass test

∞

un (x) and set D ⊂ R satisfying ∃ non-negative

Consider series of

n=1

numbers an ; n = 1, 2, · · ·

1

|un (x)|

an ∀x ∈ D; ∀n = 1, 2, · · · .

2

Series of numbers

∞

an is convergent.

n=1

∞

Then, Series

un (x) is uniformly convergent on D.

n=1

m

Proof. We have

m

m

|un (x)|

un (x)

n=k+1

an ∀x ∈ D.

n=k+1

n=k+1

∞

Since

an is convergent, it follows that

n=1

m

∀ > 0 ∃N ∈ N such that

an <

∀m, k > N .

n=k+1

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

7 / 15

Weierstrass test

∞

un (x) and set D ⊂ R satisfying ∃ non-negative

Consider series of

n=1

numbers an ; n = 1, 2, · · ·

1

|un (x)|

an ∀x ∈ D; ∀n = 1, 2, · · · .

2

Series of numbers

∞

an is convergent.

n=1

∞

Then, Series

un (x) is uniformly convergent on D.

n=1

m

Proof. We have

m

m

|un (x)|

un (x)

n=k+1

an ∀x ∈ D.

n=k+1

n=k+1

∞

Since

an is convergent, it follows that

n=1

m

∀ > 0 ∃N ∈ N such that

∀m, k > N . Therefore,

an <

n=k+1

m

∀x ∈ D; ∀ > 0 ∃N ∈ N such that |

m

un (x)|

n=k+1

an <

n=k+1

∀m, k > N . Due to Cauchy test, the series is uniformly convergent on D.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

7 / 15

Examples

∞

Ex. 1: Prove that series

n=1

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sin nx

n2 −n+1

is uniformly convergent on R.

Infinite Series and Diff. Eq.

8 / 15

Examples

∞

Ex. 1: Prove that series

We have: 1)

sin nx

n2 −n+1

Nguyen Thieu Huy (HUST)

sin nx

n2 −n+1

n=1

1

∀x

n2 −n+1

is uniformly convergent on R.

∈ R; ∀n = 1, 2, · · · .

Infinite Series and Diff. Eq.

8 / 15

Examples

∞

Ex. 1: Prove that series

We have: 1)

sin nx

n2 −n+1

sin nx

n2 −n+1

n=1

1

∀x

n2 −n+1

∞

2) Series of numbers

n=1

limn→∞

1

n2 −n+1

1

n2

is uniformly convergent on R.

∈ R; ∀n = 1, 2, · · · .

∞

1

n2 −n+1

is conv. (Comparing with

∞

= 1 and Series

n=1

n=1

1

n2

1

,

n2

where

is conv. (Riemann)).

Due to Weierstrass Test, Series is uniformly convergent on R.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

8 / 15

Examples

∞

Ex. 1: Prove that series

We have: 1)

sin nx

n2 −n+1

sin nx

n2 −n+1

n=1

1

∀x

n2 −n+1

∞

n=1

limn→∞

1

n2 −n+1

1

n2

∈ R; ∀n = 1, 2, · · · .

∞

1

2) Series of numbers

is uniformly convergent on R.

n2 −n+1

is conv. (Comparing with

∞

= 1 and Series

n=1

n=1

1

n2

1

,

n2

where

is conv. (Riemann)).

Due to Weierstrass Test, Series is uniformly convergent on R.

∞

Ex. 2: Prove that series

n=0

Nguyen Thieu Huy (HUST)

x n is uni. conv. on [− 12 , 12 ].

Infinite Series and Diff. Eq.

8 / 15

Nguyen Thieu Huy

Hanoi University of Science and Technology

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

1 / 15

Series of Functions

Definition

Let {un (x)}∞

n=1 be a sequence of functions defined ∀x ∈ D ⊂ R. Formal

sum

∞

un (x) is called a series of functions.

n=1

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

2 / 15

Series of Functions

Definition

Let {un (x)}∞

n=1 be a sequence of functions defined ∀x ∈ D ⊂ R. Formal

sum

∞

un (x) is called a series of functions.

n=1

∞

Here, when x is taken a concrete real value x0 ∈ D then

un (x0 ) is a

n=1

series of numbers.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

2 / 15

Series of Functions

Definition

Let {un (x)}∞

n=1 be a sequence of functions defined ∀x ∈ D ⊂ R. Formal

sum

∞

un (x) is called a series of functions.

n=1

∞

Here, when x is taken a concrete real value x0 ∈ D then

un (x0 ) is a

n=1

series of numbers.

∞ n

2 (x+1)n

Ex. 1:

, ∀x ∈ R, is a series of functions. Substituting x by 3,

n!

n=1

∞

we obtain

n=1

8n

n!

is a series of numbers.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

2 / 15

Series of Functions

Definition

Let {un (x)}∞

n=1 be a sequence of functions defined ∀x ∈ D ⊂ R. Formal

sum

∞

un (x) is called a series of functions.

n=1

∞

Here, when x is taken a concrete real value x0 ∈ D then

un (x0 ) is a

n=1

series of numbers.

∞ n

2 (x+1)n

Ex. 1:

, ∀x ∈ R, is a series of functions. Substituting x by 3,

n!

n=1

∞

we obtain

n=1

8n

n!

is a series of numbers.

∞

Ex. 2: Series of functions

∞

n=1

n=1

√1

n

1

nx ,

∀x ∈ R. Substituting x by 12 , we have

is a series of numbers.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

2 / 15

Domain of Convergence

Definition

∞

un (x), the set X ⊂ R is called

∞

un (x) is convergent ∀x ∈ X

n=1

domain of convergence ⇐⇒ ∞

un (x) is divergent ∀x ∈

/ X.

For series of functions

n=1

n=1

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Infinite Series and Diff. Eq.

3 / 15

Domain of Convergence

Definition

∞

un (x), the set X ⊂ R is called

∞

un (x) is convergent ∀x ∈ X

n=1

domain of convergence ⇐⇒ ∞

un (x) is divergent ∀x ∈

/ X.

For series of functions

n=1

n=1

∞

In that case, for each x ∈ X we put S(x) :=

un (x) ∈ R, then we

n=1

obtain a function x → S(x). The function S(x) is called the sum of the

∞

series of functions

∞

un (x), and we say

n=1

un (x) is convergent on X to

n=1

the function S(x).

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

3 / 15

Domain of Convergence

Definition

∞

un (x), the set X ⊂ R is called

∞

un (x) is convergent ∀x ∈ X

n=1

domain of convergence ⇐⇒ ∞

un (x) is divergent ∀x ∈

/ X.

For series of functions

n=1

n=1

∞

In that case, for each x ∈ X we put S(x) :=

un (x) ∈ R, then we

n=1

obtain a function x → S(x). The function S(x) is called the sum of the

∞

series of functions

∞

un (x), and we say

n=1

un (x) is convergent on X to

n=1

the function S(x).

∞

Ex. 1: Domain of convergence for

1

1−x

∞

=

x n is (−1, 1). Its sum is

n=0

x n;

∀x ∈ (−1, 1).

n=0

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

3 / 15

∞

Ex. 2: Domain of convergence for

∞

Its sum S(x) :=

n=1

Nguyen Thieu Huy (HUST)

n=1

1

nx

1

nx

is (1, ∞) (property of p-Series).

is called ζ-Riemann function.

Infinite Series and Diff. Eq.

4 / 15

∞

Ex. 2: Domain of convergence for

∞

Its sum S(x) :=

n=1

n=1

1

nx

1

nx

is (1, ∞) (property of p-Series).

is called ζ-Riemann function.

♣ The previous tests can be applied to find domains of convergence.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

4 / 15

∞

Ex. 2: Domain of convergence for

∞

Its sum S(x) :=

n=1

n=1

1

nx

1

nx

is (1, ∞) (property of p-Series).

is called ζ-Riemann function.

♣ The previous tests can be applied to find domains of convergence.

∞

Ex. 3: Find the Domain of Convergence for

n=1

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

xn

n ;

un (x) =

xn

n .

4 / 15

∞

Ex. 2: Domain of convergence for

∞

Its sum S(x) :=

n=1

n=1

1

nx

1

nx

is (1, ∞) (property of p-Series).

is called ζ-Riemann function.

♣ The previous tests can be applied to find domains of convergence.

∞

Ex. 3: Find the Domain of Convergence for

n=1

We have lim

n→∞

un+1 (x)

un (x)

= lim

n→∞

x n+1

n+1

xn

n

xn

n ;

|x|n

n→∞ n+1

= lim

un (x) =

xn

n .

= |x|.

Therefore, according to Ratio Test

1

If |x| < 1 then series is conv.

2

If |x| > 1 then series is div.

3

Remain to consider |x| = 1 (here Ratio Test cannot be applied)

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

4 / 15

∞

Ex. 2: Domain of convergence for

∞

Its sum S(x) :=

n=1

n=1

1

nx

1

nx

is (1, ∞) (property of p-Series).

is called ζ-Riemann function.

♣ The previous tests can be applied to find domains of convergence.

∞

Ex. 3: Find the Domain of Convergence for

n=1

We have lim

n→∞

un+1 (x)

un (x)

= lim

n→∞

x n+1

n+1

xn

n

xn

n ;

|x|n

n→∞ n+1

= lim

un (x) =

xn

n .

= |x|.

Therefore, according to Ratio Test

1

If |x| < 1 then series is conv.

2

If |x| > 1 then series is div.

3

Remain to consider |x| = 1 (here Ratio Test cannot be applied)

∞ 1

n=1 n ⇒ div. (p-Series with p = 1)

(−1)n

becomes ∞

n=1 n . This is alternating series

1

1

n ∀n and 2) limn→∞ n = 0. So, it is conv.

a) For x = 1: Series becomes

b) For x = −1: Series

1

satisfying 1) n+1

Altogether, Domain of convergence is [−1; 1).

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

4 / 15

Convergence: revisited

∞

un (x) be convergent on D to function S(x). Then, using the

Let

n=1

language ( , N) we can rewrite the notion of convergence as follows:

∞

un (x) is convergent on D to function S(x)

Series

n=1

k

⇔ ∀x ∈ D; ∀ > 0, ∃Nx; ∈ N such that |S(x) −

un (x)| <

∀k > Nx; .

n=1

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

5 / 15

Convergence: revisited

∞

un (x) be convergent on D to function S(x). Then, using the

Let

n=1

language ( , N) we can rewrite the notion of convergence as follows:

∞

un (x) is convergent on D to function S(x)

Series

n=1

k

⇔ ∀x ∈ D; ∀ > 0, ∃Nx; ∈ N such that |S(x) −

un (x)| <

∀k > Nx; .

n=1

Ex. :

1

∞

Series

x n is convergent on (−1, 1) to function

n=0

∞

2

Series

n=1

∞

1

nx

S(x) =

n=1

1

1−x

convergent on (1, ∞) to ζ-Riemann function

1

nx .

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

5 / 15

Convergence: revisited

∞

un (x) be convergent on D to function S(x). Then, using the

Let

n=1

language ( , N) we can rewrite the notion of convergence as follows:

∞

un (x) is convergent on D to function S(x)

Series

n=1

k

⇔ ∀x ∈ D; ∀ > 0, ∃Nx; ∈ N such that |S(x) −

un (x)| <

∀k > Nx; .

n=1

Ex. :

1

∞

Series

x n is convergent on (−1, 1) to function

n=0

∞

2

Series

n=1

∞

1

nx

S(x) =

n=1

1

1−x

convergent on (1, ∞) to ζ-Riemann function

1

nx .

Convergence according to the above definition is sometimes called

pointwise convergence. To do convenient calculi on a series of functions,

we need the concept of uniform convergence.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

5 / 15

Uniform convergence

Definition

∞

un (x) is called uniformly convergent on D to function S(x)

Series

n=1

⇔ ∀x ∈ D; ∀ > 0, ∃N ∈ N depending only on , not on x, such that

k

|S(x) −

un (x)| <

∀k > N .

n=1

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

6 / 15

Uniform convergence

Definition

∞

un (x) is called uniformly convergent on D to function S(x)

Series

n=1

⇔ ∀x ∈ D; ∀ > 0, ∃N ∈ N depending only on , not on x, such that

k

|S(x) −

un (x)| <

∀k > N .

n=1

We introduce the following two tests to check the uniform convergence:

Cauchy test for uniformly convergent Series of functions

∞

un (x) and set D ⊂ R satisfying

Consider series

n=1

m

∀x ∈ D; ∀ > 0 ∃N ∈ N such that |

∞

series

un (x)| <

∀m, k > N . Then,

n=k+1

un (x) is uniformly convergent on D.

n=1

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

6 / 15

Weierstrass test

∞

un (x) and set D ⊂ R satisfying ∃ non-negative

Consider series of

n=1

numbers an ; n = 1, 2, · · ·

1

|un (x)|

an ∀x ∈ D; ∀n = 1, 2, · · · .

2

Series of numbers

∞

an is convergent.

n=1

∞

Then, Series

un (x) is uniformly convergent on D.

n=1

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

7 / 15

Weierstrass test

∞

un (x) and set D ⊂ R satisfying ∃ non-negative

Consider series of

n=1

numbers an ; n = 1, 2, · · ·

1

|un (x)|

an ∀x ∈ D; ∀n = 1, 2, · · · .

2

Series of numbers

∞

an is convergent.

n=1

∞

Then, Series

un (x) is uniformly convergent on D.

n=1

m

Proof. We have

m

m

|un (x)|

un (x)

n=k+1

an ∀x ∈ D.

n=k+1

n=k+1

∞

Since

an is convergent, it follows that

n=1

m

∀ > 0 ∃N ∈ N such that

an <

∀m, k > N .

n=k+1

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

7 / 15

Weierstrass test

∞

un (x) and set D ⊂ R satisfying ∃ non-negative

Consider series of

n=1

numbers an ; n = 1, 2, · · ·

1

|un (x)|

an ∀x ∈ D; ∀n = 1, 2, · · · .

2

Series of numbers

∞

an is convergent.

n=1

∞

Then, Series

un (x) is uniformly convergent on D.

n=1

m

Proof. We have

m

m

|un (x)|

un (x)

n=k+1

an ∀x ∈ D.

n=k+1

n=k+1

∞

Since

an is convergent, it follows that

n=1

m

∀ > 0 ∃N ∈ N such that

∀m, k > N . Therefore,

an <

n=k+1

m

∀x ∈ D; ∀ > 0 ∃N ∈ N such that |

m

un (x)|

n=k+1

an <

n=k+1

∀m, k > N . Due to Cauchy test, the series is uniformly convergent on D.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

7 / 15

Examples

∞

Ex. 1: Prove that series

n=1

Nguyen Thieu Huy (HUST)

sin nx

n2 −n+1

is uniformly convergent on R.

Infinite Series and Diff. Eq.

8 / 15

Examples

∞

Ex. 1: Prove that series

We have: 1)

sin nx

n2 −n+1

Nguyen Thieu Huy (HUST)

sin nx

n2 −n+1

n=1

1

∀x

n2 −n+1

is uniformly convergent on R.

∈ R; ∀n = 1, 2, · · · .

Infinite Series and Diff. Eq.

8 / 15

Examples

∞

Ex. 1: Prove that series

We have: 1)

sin nx

n2 −n+1

sin nx

n2 −n+1

n=1

1

∀x

n2 −n+1

∞

2) Series of numbers

n=1

limn→∞

1

n2 −n+1

1

n2

is uniformly convergent on R.

∈ R; ∀n = 1, 2, · · · .

∞

1

n2 −n+1

is conv. (Comparing with

∞

= 1 and Series

n=1

n=1

1

n2

1

,

n2

where

is conv. (Riemann)).

Due to Weierstrass Test, Series is uniformly convergent on R.

Nguyen Thieu Huy (HUST)

Infinite Series and Diff. Eq.

8 / 15

Examples

∞

Ex. 1: Prove that series

We have: 1)

sin nx

n2 −n+1

sin nx

n2 −n+1

n=1

1

∀x

n2 −n+1

∞

n=1

limn→∞

1

n2 −n+1

1

n2

∈ R; ∀n = 1, 2, · · · .

∞

1

2) Series of numbers

is uniformly convergent on R.

n2 −n+1

is conv. (Comparing with

∞

= 1 and Series

n=1

n=1

1

n2

1

,

n2

where

is conv. (Riemann)).

Due to Weierstrass Test, Series is uniformly convergent on R.

∞

Ex. 2: Prove that series

n=0

Nguyen Thieu Huy (HUST)

x n is uni. conv. on [− 12 , 12 ].

Infinite Series and Diff. Eq.

8 / 15

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