Theorem For Uniform Convergence Of Power Series

Theorem For Uniform Convergence Of Power Series :

          The power series Σ aₙxⁿ is uniformly convergent for |x| ≤P≤R where R is the radius of convergence .

Proof :

          Consider a number ρ' between ρ and R . Since , the series is convergent for |x| = ρ' , then by definition there exists k 

 independent of n so that 

               |aₙPⁿ|< k ∀ n 

     ⇒ for |x| ≤ ρ , |aₙxⁿ| = 

                           |aₙρⁿ(x/ρ')ⁿ|<(ρ/ρ')ⁿ

which is independent of x .

       But the series is geometric series with common ratio ρ/ρ' < 1 , therefore the series 

          kΣ (ρ/ρ')ⁿ is convergent .

      Thus , by Weirstrass's M_test the power series is uniformly convergent for
 |x| <ρ<R .

    Hence , every power series is uniformly convergent within its radius of convergence .