Sequence and Series of Functions

Sequence of functions

• Definition:A sequence of functions is simply a set of functions un(x), n = 1, 2, . . . defined on a common domain D.

• A frequently used example will be the sequence of functions {1, x, x2, . . .}, x ϵ [-1, 1]

Sequence of Functions Convergence

• Let D be a subset of and let {un} be a sequence of real valued functions defined on D. Then {un} converges on D to g if

for each x ϵ D• More formally, we write that

if given any x ϵ D and given any > 0, there exists a natural number N = N(x, ) such that

lim nnu x g x

lim nnu g

,nu x g x n N

Sequence of Functions Convergence

• Example 1Let {un} be the sequence of functions on defined by un(x) = nx.

This sequence does not converge on because for any x > 0 lim n

nu x

Sequence of Functions Convergence

• Example 2: Consider the sequence of functions

The limits depends on the value of xWe consider two cases, x = 0 and x 01. x = 0 2. x 0

1, , 1, 2, 3,....

1nu x x nnx

lim 0 lim1 1nn nu

1lim lim 0

1nn nu x

nx

Sequence of Functions Convergence

Therefore, we can say that {un} converges to g for |x| < , where

0, 0

1, 0

xg x

x

Sequence of Functions Convergence

• Example 3:Consider the sequence {un} of functions defined by

Show that {un} converges for all x in

2

2, for all in n

nx xu x x

n

Sequence of Functions Convergence

• SolutionFor every real number x, we have

Thus, {un} converges to the zero function on

2

22 2

1 1lim lim lim lim 0 0 0nn n n n

x xu x x x

n n n n

Sequence of Functions Convergence

• Example 4:Consider the sequence {un} of functions defined by

Show that {un} converges for all x in

sin 3, for all in

1n

nxu x x

n

Sequence of Functions Convergence

• SolutionFor every real number x, we have

Moreover,

Applying the squeeze theorem, we obtain that

Therefore, {un} converges to the zero function on

1 sin( 3) 1

1 1 1

nx

n n n

1

lim 01n n

lim 0, for all in nnu x x

Sequence of Functions Convergence

• Example 5:Periksalah kekonvergenan barisan fungsi pada himpunan bilangan realSolution:Akan ditinjau untuk beberapa kasus:1. |x| < 1 2. |x| > 1 tidak ada3. x = 1 4. x = -1 tidak ada

Barisan tersebut konvergen untuk 1 < x ≤ 1

1 nx

lim 1 1n

nx

lim 1 n

nx

lim 1 0n

nx

lim 1 n

nx

Sequence of Functions Convergence

• Example 6Consider the sequence {fn} of functions defined byWe recall that the definition for convergence suggests that for each x we seek an N such that .This is not at first easy to see. So, we will provide some simple examples showing how N can depend on both x and

, [0,1], 1,2,....nnf x x x n

,nf x g x n N

Sequence of Functions Convergence

Sequence of Functions Convergence

Uniform Convergence

• Let D be a subset of and let {un} be a sequence of real valued functions defined on D. Then {un} converges uniformly on D to g if given any > 0, there exists a natural number N = N() such that

, andnu x g x n N x D

Uniform Convergence

• Example 7:Ujilah konvergensi uniform dari example 5a. pada interval -½ < x < ½ b. pada interval -1 < x < 1

Series of Functions

• Definition:An infinite series of functions is given by x ϵ D.

1

,nn

u x

Series of Functions Convergence

• is said to be convergent on D if the sequence of partial sums {Sn(x)}, n = 1, 2, ...., where is convergent on D

• In such case we write and call S(x) the sum of the series

• More formally,if given any x ϵ D and given any > 0, there exists a natural number N = N(x, ) such that

ju x

1

N

N nn

S x f x

lim n

nS x S x

,nS x S x n N

Series of Functions Convergence

• If N depends only on and not on x, the series is called uniformly convergent on D.

Series of Functions Convergence

• Example 8:Find the domain of convergence of (1 – x) + x(1 – x) + x2(1 – x) + ....

Series of Functions Convergence

• Example 9:Investigate the uniform convergence of

2 2 2

222 2 2

... ...1 1 1

n

x x xx

x x x

Exercise

1. Consider the sequence {fn} of functions defined by for 0 ≤ x ≤ 1. Determine whether {fn} is convergent.

2. Let {fn} be the sequence of functions defined by for /2 ≤ x ≤ /2. Determine the convergence of the sequence.

3. Consider the sequence {fn} of functions defined by on [0, 1]Show that {fn} converges to the zero function

2 nnf x n x

cosnnf x x

1n

nf x nx x

Exercise

4. Find the domain of convergence of the series a) b)

c) d)

e)

5. Prove that converges for -1 ≤ x < 1

31

n

n

x

n

1

1 1

2 3 1

n n

nn

x

n

21

1

1n

n n x

2

1

1

1

n

n

xn

x

21 1

nx

n

e

n n

1

1.3.5...(2 1)

2.4.6...(2 )n

n

nx

n

Exercise

6. Investigate the uniform convergence of the series

7. Let Prove that {fn} converges but not uniformly on (0, 1)

1 [1 1 ][1 ]n

x

n x nx

1, 0 1, 1,2,3,...

1nf x x nnx