Transcript Sequence and Series of Functions

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
lim un  x   g  x 
n 
for each x ϵ D
• More formally, we write that
lim un  g
n 
if given any x ϵ D and given any  > 0, there exists
a natural number N = N(x,  ) such that
un  x   g  x    , n  N
Sequence of Functions Convergence
• Example 1
Let {un} be the sequence of functions on 
defined by un(x) = nx.
This sequence does not converge on 
un  x    for any x > 0
because lim
n 
Sequence of Functions Convergence
• Example 2: Consider the sequence of
functions
1
un  x  
,
1  nx
x  , n  1, 2, 3,....
The limits depends on the value of x
We consider two cases, x = 0 and x  0
un  0   lim1  1
1. x = 0  lim
n 
n 
1
2. x  0  lim un  x   lim
0
n 
n 
1  nx
Sequence of Functions Convergence
Therefore, we can say that {un} converges to g
for |x| < , where
0, x  0
g  x  
1, x  0
Sequence of Functions Convergence
• Example 3:
Consider the sequence {un} of functions
defined by
nx  x 2
un  x  
, for all x in 
2
n
Show that {un} converges for all x in 
Sequence of Functions Convergence
• Solution
For every real number x, we have
x x2
1 2
1

lim un  x   lim  2  x  lim   x  lim 2   0  0  0
n
n n
n
 n n 
 n n 
Thus, {un} converges to the zero function on

Sequence of Functions Convergence
• Example 4:
Consider the sequence {un} of functions
defined by
sin  nx  3
un  x  
, for all x in 
n 1
Show that {un} converges for all x in 
Sequence of Functions Convergence
• Solution
For every real number x, we have
Moreover,
1
sin(nx  3)
1


n 1
n 1
n 1
1
lim
0
n 
n 1
Applying the squeeze theorem, we obtain that
lim un  x   0, for all x in 
n 
Therefore, {un} converges to the zero function on 
Sequence of Functions Convergence
• Example 5:
Periksalah kekonvergenan barisan fungsi 1  x n 
pada himpunan bilangan real
Solution:
Akan ditinjau untuk beberapa kasus:
1  xn   1
1. |x| < 1  lim

n 
2. |x| > 1  lim 1  x n  tidak ada
n
3. x = 1  lim 1  x n   0
n 
4. x = -1  lim 1  x n  tidak ada
n
Barisan tersebut konvergen untuk 1 < x ≤ 1
Sequence of Functions Convergence
• Example 6
Consider the sequence {fn} of functions defined
by fn  x  xn , x [0,1], n  1,2,....
We recall that the definition for convergence
suggests that for each x we seek an N such that
f n  x   g  x    , n  N .
This is not at first easy to see.
So, we will provide some simple examples
showing how N can depend on both x and 
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
un  x   g  x    , n  N and x  D
Uniform Convergence
• Example 7:
Ujilah konvergensi uniform dari example 5
a. pada interval -½ < x < ½
b. pada interval -1 < x < 1
Series of Functions
• Definition:
An infinite series of functions is given by

 un  x  , x ϵ D.
n 1
Series of Functions Convergence
•  u j  x  is said to be convergent on D if the
sequence of partial
sums
{S
(x)},
n
=
1,
2,
....,
n
N
where SN  x    fn  x  is convergent on D
n 1
Sn  x   S  x  and call
• In such case we write lim
n 
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
S n  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
x
x
x2 

2
2
1 x
1 x


2
 ... 
x2
1  x 
2
n
 ...
Exercise
1. Consider the sequence {fn} of functions
2 n
defined by fn  x   n x for 0 ≤ x ≤ 1.
Determine whether {fn} is convergent.
2. Let {fn} be the sequence of functions defined
n
by fn  x   cos  x  for /2 ≤ x ≤ /2.
Determine the convergence of the sequence.
3. Consider the sequence {fn} of functions
n
defined by f n  x   nx 1  x  on [0, 1]
Show that {fn} converges to the zero function
Exercise
4. Find the domain of convergence
of the series
n
n


 1  x  1
xn
a)  3
b)  2n 3n  1


n 1
n 1 n

1
c)  n 1  x
n 1
e)
5.

2

n
d)
 1 x 
n 


1

x


n 1

n
2

enx

2
n
 n 1
n 1
1.3.5...(2n  1) n
Prove that  2.4.6...(2n) x
n 1

-1 ≤ x < 1
converges for
Exercise
6. Investigate the uniform convergence of the
series 
x
 [1   n 1 x][1  nx]
n 1
fn  x  
1
, 0  x  1, n  1, 2,3,...
1  nx
7. Let
Prove that {fn} converges but not uniformly on
(0, 1)