Mathquery

Abel's Theorem For Power Series : Statement :                                  ∞            If the series Σ aₙ is convergent and                                  n=0               ∞ has the sum s , then the series Σ aₙ xⁿ is                                                       n=0 uniformly convergent for 0≤x≤1 and                                   ∞                      lim      Σ aₙ xⁿ = s .                     x-->1  n=0 Proof :              Given that the series Σ aₙ is convergent , therefore we have for n≥m        |aₙ₊₁+ aₙ₊₂ + .......+aₙ₊ₚ| <ε , for every integral value of p>0 .    Also , since the sequence <xₙ> is monotonic  decreasing for all values of  x∈ [0,1] .  Then from Abel's inequality   |aₙxⁿ + aₙ₊₁xⁿ⁺¹j +.....aₙ₊ₚxⁿ⁺ᵖ|≤εx' ≤ ε                                                 (x∈[0,1])  Therefore , the series Σ aₙxⁿ is uniformly convergent for 0≤x≤1 . which implies  Σ

Theorem For Uniform Convergence Of Power Series :           T he 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 .