Radius Of Convergence

             Radius Of Convergence

     The number R is called the radius of convergence of the power series 

   ∞

   Σ aₙ xⁿ and the set of all x for which 

  n=0

  |x|< R i.e the open interval ]-R,R[ is called interval of convergence .

         A power series Σ aₙ xⁿ absolutely converges for values of x inside the circle of convergence and diverges outside the circle . For values of x on the circumference of the circle , the series may converge , diverge or oscillate . 

For Example :

                       ∞

• The series Σ  aₙ xⁿ converges for |x|< 1   

                      n=0

and diverges for |x| < 1 and diverges for |x|≥1 .

                          ∞

• The series     Σ  xⁿ/n converges for -1≤x≤1

                         n=0

and diverges else where .

          

• The series Σ xⁿ / n² converges absolutely for |x| ≤ 1 and diverges for |x|>1 .

Expression Of The Radius Of Convergence :

         Therefore , the radius of convergence can also be found by the relation ,

            R = lim  |aₙ /aₙ₊₁ |

                n--->∞

Provided the limit exist . We can also find the radius of convergence by using the following formula ,

             R =  1 / lim  sup ⁿ√|aₙ| 

                         n-->∞