Calculus 3 2 giải tích

2: Domain of convergence forn x is called ζ-Riemann function.. Radius and Interval of of ConvergenceThen, the interval −R, R is called the interval of convergence... Radius and Interval

Infinite Series and Differential Equations

Nguyen Thieu Huy

Hanoi University of Science and Technology

un(x ) is called a series of functions.

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

∞

P

n=1

un(x0) is aseries of numbers

un(x ) is called a series of functions.

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

∞

P

n=1

un(x0) is aseries of numbers

un(x ) is called a series of functions.

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

∞

P

n=1

un(x0) is aseries of numbers

un(x ) is called a series of functions.

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

∞

P

n=1

un(x0) is aseries of numbers

Ex 1: Domain of convergence for

Ex 1: Domain of convergence for

Ex 1: Domain of convergence for

Ex 2: Domain of convergence for

n x is called ζ-Riemann function

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

Ex 3: Find the Domain of Convergence for

x n+1 n+1

x n n

= lim

n→∞

|x|n n+1 = |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)

a) For x = 1: Series becomes ∞n=11n ⇒ div (p-Series with p = 1)

b) For x = −1: Series becomes P∞

n=1

(−1) n

n This is alternating seriessatisfying 1) n+11 6 1n ∀n and 2) limn→∞1n = 0 So, it is conv.Altogether, Domain of convergence is [−1; 1)

Ex 2: Domain of convergence for

n x is called ζ-Riemann function

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

Ex 3: Find the Domain of Convergence for

x n+1 n+1

x n n

= lim

n→∞

|x|n n+1 = |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)

a) For x = 1: Series becomes ∞n=11n ⇒ div (p-Series with p = 1)

b) For x = −1: Series becomes P∞

n=1

(−1) n

n This is alternating seriessatisfying 1) n+11 6 1n ∀n and 2) limn→∞ 1n = 0 So, it is conv.Altogether, Domain of convergence is [−1; 1)

Ex 2: Domain of convergence for

n x is called ζ-Riemann function

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

Ex 3: Find the Domain of Convergence for

x n+1 n+1

x n n

(or ρ := lim

With the conventions 10 = ∞ and ∞1 = 0, we can write R = 1ρ

Note on Domain of Conv.: To calculate the domain of conv for P anxn

we can use one of the following two methods:

1 To compute as previously, or

2 To compute the radius of conv R, and then interval of conv

(−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

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

Calculation of radius of convergence

Theorem

For P anxn with ρ := lim

n→∞

a n+1

a n

(or ρ := lim

With the conventions 10 = ∞ and ∞1 = 0, we can write R = 1ρ

Note on Domain of Conv.: To calculate the domain of conv for P anxn

we can use one of the following two methods:

1 To compute as previously, or

2 To 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

Theorem

For P anxn with ρ := lim

n→∞

a n+1

a n

(or ρ := lim

With the conventions 10 = ∞ and ∞1 = 0, we can write R = 1ρ

Note on Domain of Conv.: To calculate the domain of conv for P anxn

we can use one of the following two methods:

1 To compute as previously, or

2 To 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 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

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

... class="text_page_counter">Trang 22

1 n2−n+1 n2

Due to Weierstrass Test, Series is uniformly convergent on R

Ex 2: Prove that series... class="text_page_counter">Trang 23 < /span>

1 n2−n+1 n2

Due to Weierstrass Test, Series is uniformly convergent on R

Ex 2: Prove that series... class="text_page_counter">Trang 26

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

Ex 2: Prove that series

2 n is