Answer to Question #304169 in Calculus for itz_perfect

Question #304169

Write down T3(x), T4(x), and T5(x) for the Taylor series of f(x) = ln (3 + 4x) about x = 0.

Graph all three of the Taylor polynomials and f(x) on the same graph for the interval [−1/2,2]using programming tool.

Expert's answer

"\\ln(1+x)=\\Sigma_{n=1}^{\\infin}(-1)^{n-1}\\frac{x^n}{n}\\\\\n\\ln(3+4x)=ln(3)+ln(1+(\\frac{4}{3}x))=ln(3)+\\Sigma_{n=1}^{\\infin}(-1)^{n-1}\\frac{(\\frac{4}{3}x)^n}{n}=\\\\\n=ln(3)+\\frac{4}{3}x-\\frac{1}{2}(\\frac{4}{3})^2x^2+\\frac{1}{3}(\\frac{4}{3})^3x^3-\\frac{1}{4}(\\frac{4}{3})^4x^4+\\frac{1}{5}(\\frac{4}{3})^5x^5+O(x^6)\\\\\nT_3(x)=ln(3)+\\frac{4}{3}x-\\frac{8}{9}x^2+\\frac{64}{81}x^3\\\\\nT_4(x)=ln(3)+\\frac{4}{3}x-\\frac{8}{9}x^2+\\frac{64}{81}x^3-\\frac{64}{81}x^4\\\\\nT_5(x)=ln(3)+\\frac{4}{3}x-\\frac{8}{9}x^2+\\frac{64}{81}x^3-\\frac{64}{81}x^4+\\frac{1024}{1215}x^5\\\\"

convergence interval"(-\\frac{3}{4},\\frac{3}{4}]"