This week, we’ll work out some Taylor Series expansions of popular functions. • f(x) = 1/(1-x) • f(x) = e^x • f(x) = ln(1 + x) 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 a R-Markdown document.
1- Taylor Series expansions for: \[f({\bf x}) =\frac{1}{(1-x)}\]
When x = 0, f(0) = 1 also when x=0 successive derivatives are first Derivative of f(x)
\[{f'(x)} =(-1)(-1) (1-x)^{-2} =1\] Second Derivative of f(x)
\[{f''(x)} =(-1^2)(-1)(-2) (1-x)^{-3} =2\] Third Derivative of f(x)
\[{f'''(x)} =(-1^3)(-1)(-2)(-3) (1-x)^{-4} =6\]
nth derivative of f(x) is
\[{f^{n} (x)} =(n!) (1-x)^{-n} \]
Taylor Series expansions for: \[\frac{1}{(1-x)}\] when a=0 \[f({\bf x}) =\sum_{n=0}^{\infty} \frac{f^n(a)}{n!} (x-a)^n \] \[f({\bf x}) = 1 + x + x^2 + x^3 + x^4 + \ldots \]
2- Taylor Series expansions for: \(f({\bf x}) =e^{x}\)
When x = 0, f(0) = 1 also when x=0 successive derivatives are \(e^{x}\)=1
Taylor Series expansions for: \(f({\bf x}) =e^{x}\) when a=0 \[f({\bf x}) =\sum_{n=0}^{\infty} \frac{f^n(a)}{n!} (x-a)^n \] \[f({\bf x}) = 1 + x + \frac{x^2} {2!} + \frac{x^3} {3!} + \frac{x^4} {4!} + \ldots \]
3- Taylor Series expansions for: \(f({\bf x}) =ln(1 + x)\)
When x = 0, f(0) = 0
first Derivative of f(x) \({f'(x)} =\frac{1}{(1+x)} =1\)
Second Derivative of f(x) \[{f''(x)} =(-1) (1+x)^{-2} =-1\]
Third Derivative of f(x) \[{f'''(x)} =(-1)(-2) (1+x)^{-3} =2\]
Taylor Series expansions for: \(f({\bf x}) =ln(1 + x)\) when a=0 \[f({\bf x}) =\sum_{n=0}^{\infty} \frac{f^n(a)}{n!} (x-a)^n \] \[ =\sum_{n=0}^{\infty} \frac{(-1)^n }{n+1} {x^{(n+1)}}\] \[f({\bf x}) = x - \frac{x^2} {2} + \frac{x^3} {3} - \frac{x^4} {4} + \ldots \]