Sequence and Series of Functions. Sequence of functions Definition: A sequence of functions is...
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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