Question

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.

Answer 1

$\ln(1+x)=\Sigma_{n=1}^{\infin}(-1)^{n-1}\frac{x^n}{n}\\ \ln(3+4x)=ln(3)+ln(1+(\frac{4}{3}x))=ln(3)+\Sigma_{n=1}^{\infin}(-1)^{n-1}\frac{(\frac{4}{3}x)^n}{n}=\\ =ln(3)+\frac{4}{3}x-\frac{1}{2}(\frac{4}{3})^2x^2+\frac{1}{3}(\frac{4}{3})^3x^3-\frac{1}{4}(\frac{4}{3})^4x^4+\frac{1}{5}(\frac{4}{3})^5x^5+O(x^6)\\ T_3(x)=ln(3)+\frac{4}{3}x-\frac{8}{9}x^2+\frac{64}{81}x^3\\ T_4(x)=ln(3)+\frac{4}{3}x-\frac{8}{9}x^2+\frac{64}{81}x^3-\frac{64}{81}x^4\\ T_5(x)=ln(3)+\frac{4}{3}x-\frac{8}{9}x^2+\frac{64}{81}x^3-\frac{64}{81}x^4+\frac{1024}{1215}x^5\\$

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