Taylor's Theorem For Power Series
Taylor's Theorem For Power Series : Statement : Let ∞ Σ aₙ xⁿ be a power series with n= 0 radius of convergence R , and let ∞ f(x) = Σ aₙ xⁿ , |x| < R n=0 Then for any a∈ ]-R ,R[ , prove that f can be expanded in a power series about 'a' which converges for |x-a| < R- |a| , and ∞ f(x) = Σ f⁽ⁿ⁾(a) (x-a)ⁿ /n! , |x-a|<R-|a| n=0 Proof : Suppose |x-a|<R-|a|. Then |x|≤|x-a|+|a|<R and thus Σaₙxⁿ converges . ∞ ∞ Now , f(x) = Σ aₙxⁿ = Σ aₙ(x-a + a)ⁿ n=0 n=0 ∞ n = Σ aₙ Σ ⁿCₘ aⁿ⁻ᵐ(x-a)ᵐ ......(1) n=0 m=0 We wish to change the order of summation in this express