Mathquery

              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 :      

Power Series And It's Theorems : Definition Of Power Series :                 A series of the form      ∞      Σ aₙ(x-x₀)ⁿ = a₀+a₁(x-x₀)+a₂(x-x₀)²+......     n=0                                            + aₙ(x-x₀)ⁿ+.... is called a power series .      where x is a continuous variable and the constants aₙ,x₀ are real and independent of x . Remarks :    --> If we change the variable x= t+x₀ ,              then the power series reduces to                 ∞                 Σ aₙ tⁿ  .                 n=0    -->For x=0 , every power series is                      convergent , whatever the value of              co_efficients .  --> A power series is either        (1) convergent for no value of x other              than x= 0 , then it is said to be                      nowhere convergent .        (2) convergent for all values of x and                is called everywhere conver