1 This week, we’ll work out some Taylor Series expansions of popular functions. f (x) = (1−x) f (x) = ex f (x) = ln(1 + x) f(x)=x(1/2) For each function, only consider its valid ranges as indicated in the notes when you are computing the Taylor Series expansion. Please submit your assignment as an R- Markdown document.

Taylor Series

\[f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(c)}{n!}(x-c)^n \]

  1. \[f(x)=\frac{1}{(1-x)}\] Then this is the dericatives example: \[f^{(0)}(c)=\frac{1}{(1-c)}\] \[f^{(1)}(c)=\frac{1}{(1-c)^{(2)}}\] \[f^{(2)}(c)=\frac{1}{(1-c)^{(3)}}\] \[f^{(3)}(c)=\frac{1}{(1-c)^{(4)}}\] \[f(x)=\sum_{n=0}^{\infty}\frac{1}{(1-c)^{(n+1)}}(x-c)^n \] if the c = 0 it would be

\[1+x+x^2+x^3+...\] 2. \[f(x)=e^x\] the dericatives is always

\[f^{(0)}(c)=e^c\] \[f^{(1)}(c)=e^c\] \[f^{(2)}(c)=e^c\] \[f(x)=\sum_{n=0}^{\infty}\frac{(x-c)^n}{n!} \]

if the c = 0 it would be \[1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...\] 3. \[f(x)=ln(1+x)\] \[f^{(0)}(c)=ln(1+c)\] \[f^{(1)}(c)=\frac{1}{(c+1)}\] \[f^{(2)}(c)=\frac{1}{(c+1)^{2}}\] \[f^{(3)}(c)=\frac{1}{(c+1)^{3}}\] I am not sure the correct way to do the next step after that.

  1. \[f(x)=x^{(1/2)}\] if c = 0 then it is 0 for all.