infinite series : Introduction

                    INFINITE SERIES                                              

Definition  :  

                    If { u_n }  is a sequence of real numbers and  S_n = u₁+u₂+u₃+…+u_n ,for some +ve integer n, then the sequence  { s_n } is called an infinite series.

   The number u_n is called the n^th term of the series.

   The number s_n is called the n^th partial sum of the series.

The infinite series { s_n } is denoted by Σ u_n = u₁+u₂+u₃+…

Convergence of Series  :                         

                                             Let Σ u_n be a series of real numbers with partial sums

                            s_n = s₁+s₂+…+s_{n .}

        * If the sequence  { s_n } converges to s, we may say that the series Σ u_n converges to s.

        *The number s is called the sum of the series and we write Σu_n =s.

        *If the limit of the sequence {s_n } does not exist we say that the series Σu_n diverges.

Note  :

  *         If Σu_n converges to s , then s_n ≤ s ∀ n.                                        

           *         If Σu_n diverges then for any real number M>0 there exists m >0 such that 

                      s_n > M for n ≥m      

  If Σu_n is finite then we say the infinite series Σu_n convergent otherwise the series Σu_n is             divergent.

         For example  :  

    * Consider the series Σ 1/2ⁿ = 1+ ½ + ½² +… = Σ u_n

        Here u_n = 1/2ⁿ and s_n = 1+ ½ + ½² +…+1/2ⁿ

       This is a Geometric Progression with a=1 and r=1/2. 

     ∴ s_n = a(1-rⁿ) / (1-r)                                                                                       Back

             =  1 ( 1-1/2ⁿ⁺¹) )/( 1-1/2)

             = 2( 1-1/2ⁿ⁺¹⁾                                                                                                              

                =2- 1/2ⁿ

∴ lim s_n =2 and hence Σu_n = Σ 1/2ⁿ  is convergent.

   If Σs_n = ∞ then the series is divergent to ∞.

   For example  : 

          consider the series   Σk = k+k+k+… diverges to ∞.

    * The series  Σn= 1+2+3+…. Is diverges.

·        * The series Σ (-1)ⁿ =-1+1-1+1-1+1… is diverges.