TAYLOR SERIES is definied as \[f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}c}{n!}(x-c)^n \]
Function 1: \(f(x) = \frac{1}{(1-x)}\)
\(f(c) + f^1(c)(x-c) + \frac{f^2}{2!}(c)(x-c) + ...\)
The derivatives are: \(f(c) = \frac{1}{1-c}, f(0) = 1\) \(f'(c) = \frac{1}{(1-c)^2}, f^1(0) = 1\) \(f''(c) = \frac{2}{(1-c)^3}, f^2(0) = 2\)
Adding the expressions into the Taylor Series Expansion (when x=0): \(f(c) + f'(c)(x-c) + \frac{f''}{2!}(x-c)+ ...\) \(=1 + 1x + \frac{2}{2!}x^2\)
The valid range for this series is (−1,1).
Function 2: \(f(x) = e^x\)
\(e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}\)
The derivatives are: $f^0(c) = e^c,f(0) = 1 $ $f’(c) = e^c,f’(0) = 1 $ $f’‘(c) = e^c,f’’(0) = 1 $
Adding the expressions into the Taylor Series Polynomial (when x=0): \(\frac{e^c}{0!}(x-c)^0 + \frac{e^c}{1!}(x-c)^1+ \frac{e^c}{2!}(x-c)^2+ ...\) \(e^c + e^c(x-c)+e^c\frac{(x-c)^2}{2!}+...\) \(=1 + x + \frac{x^2}{2}+ ...\)
Function 3: \(f(x) = ln(1+x)\)
\(ln(1+x) = -\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^n}{n}\)
The derivatives are: \(f(c) = ln(1 + c), f(0) = 0\) \(f'(c) = \frac{1}{1+c}, f'(0) = 1\) \(f''(c) = \frac{-1}{(1+a)^2}, f''(0) = -1\)
Adding the expressions into the Taylor Series Polynomial (when x=0): \(f(c) + f'(c)(x-c) + \frac{f''}{2!}(x-c)+..\) \(=x-\frac{x^2}{2}+\frac{x^3}{3}+...\)