SEQUENCE AND SERIES FUNCTION

 Sequence 

       A sequence is a function from  N→ R so far we consider sequence and series whose terms where numbers .

Now we shall consider swquence and series whise terms depend on a variables i.e. those whose terms are real vector function defined on an interval at the domain . Thus a sequence of function is a function from N to the set of real valued function defined on a set S, where S is subset of  R.

We denote the terms of sequnece by  fn(x) and sequence by { fn} 

ex.,   fn:S→ R    S = [0 , 1]  ,    fn(x) = x/n , x ∈ [0 , 1]

n = 1  ⇒  f1(x) = x/1  = x

n = 2    ⇒   f2(x),   = x/2 

n = 3    ⇒  f3(x)  = x/3

n = m   ⇒  fm(x) =  x/m

Pointwise convergence of a sequence of function 

     A sequence { fn}  of real valued function an a non empty subset S of R is said to be converges pointwies to a function f on S . If given ɛ > 0 for each x ∈S   ∃ n0 ∈ N  s.t.                |fn(x) – f(x)| < ε  ∀ n > n0   where f : S→ R and we write  fn → f  .

Uniform converges of a sequence of function

  A sequence { fn (x)} be a sequence of function defined on a set S ⊂ R we say that  { fn (x)}  conerges to f(x) uniformaly if for any  ɛ > 0 for each x ∈S   ∃ n0 ∈ N  s.t.          |fn(x) – f(x)| < ε  ∀ n ≽ n0 , ∀ x ∈ S and we write  fn→ f unoformaly .

Alternatively, we can define the uniform convergence of a sequence of functions, as follows.                                                                                                                                        A sequence of functions fn(x); n = 1, 2, 3,…. Is said to be converges uniformly to f if and only if;

 

That means, supx∈E |fn(x) – f(x)| → 0 as n → ∞.

Series 

  Let (x) be a sequence in R 

let  ,     S1 = x1

             S2 = x1+ x2

             S3 = x1 +x2 + x3

               S n = x1 +x2 + x3 .... + x n

where   S n = x1 +x2 + x3 .... + x n               is called nth partial sum . Then the sequence of partial sum ( Sn ) is called a series . 

Notation : Σ xn

Series of Function :

  Let E ≠ ф  , E⊆ R let  { fn}  be a sequence of functions defined on E .

 let.,   S1(x) =f1(x)

          S2 (x) =  f1 x1+f2 ( x)

           S3 (x) = f1 (x) +f2 (x) + f3 (x)

             S n (x) =  f1 (x) +f2 (x) + f3 (x)+ .... +fn(x)

The sequence of partial sum { S n} is defined to be the series of function {fn} on E 

Pointwise converges of a series of function : 

        Let   E ≠ ф  , E⊆ R let  { fn}  be a sequence of functions defined on E . Σ xn  

is said to be a series converges pointwise to a function f on E . If the sequence of partial sum { S n} converges pointwise to f on E . For every ε > 0 and for each x ∈ E , ∃ n0 ∈ N  s.t. | S n(x) – s(x)| < ε  ∀ n ≽ n0  .

Uniform converges of a series of function :

A series of functions ∑fn(x); n = 1, 2, 3,… is said to be uniformly convergent on E if the sequence {Sn} of partial sums defined by

∑�=1���(�)=��(�)

Alternatively, we can define the uniform convergence of a series as follows.

Suppose gn(x) : E → ℝ is a sequence of functions, we can say that the series

∑�=1∞��(�)

converges uniformly to S(x) on E if and only if the partial sum

converges uniformly to S(x) on E.

��(�)=∑�=1���(�)








converges uniformly to S(x) on E.



























































lim�→∞(sup���|��(�)−�(�)|)=0




That means, supx∈E |fn(x) – f(x)| → 0 as n → ∞.














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