Taylor Series Expansion
\[f(x) = \sum_{j=0}^{\infty} \frac{f^{(j)}(a)}{j!}(x-a)^j\]
\[f(x) = \frac{f(a)}{0!} + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \frac{f^{(4)}(a)}{4!}(x-a)^4 + ...\]
We’ll work out some Taylor Series expansions of popular functions
\[f(x) = \frac{1}{(1-x)}\]
at \(a = 0\)
\[f'(x) = \frac{1}{(1-x)^2} >> f'(0) = \frac{1}{(1-0)^2} = 1\] \[f''(x) = 2* \frac{1}{(1-x)^3} >> f''(0) = 2 * \frac{1}{(1-0)^3} = 2\] \[f'''(x) = 3*2* \frac{1}{(1-x)^4} >> f''';(0) = 3 * 2 * \frac{1}{(1-0)^4} = 6\] \[f^{(4)}(x) = 4*3*2* \frac{1}{(1-x)^5} >> f^{(4)}(0) = 4 * 3 * 2 * \frac{1}{(1-0)^4} = 24\] \[f(x) = \frac{f(0)}{0!} + \frac{1(0)}{1!}(x-0) + \frac{2(0)}{2!}(x-0)^2 + \frac{6(0)}{3!}(x-0)^3 + \frac{24(0)}{4!}(x-0)^4 + ...\]
\[f(x) = 1 + x + x^2 + x^3 + x^4 + ...\]
\[f(x) = e^x\]
at \(a = 0\)
\[f'(x) = e^2 >> f'(0) = 1\] \[f''(x) = e^2 >> f''(0) = 1\] \[f'''(x) = e^2 >> f'''(0) = 1\] \[f^{(4)}(x) = e^2 >> f^{(4)}(0) = 1\] \[f(x) = \frac{f(0)}{0!} + \frac{1(0)}{1!}(x-0) + \frac{1(0)}{2!}(x-0)^2 + \frac{1(0)}{3!}(x-0)^3 + \frac{1(0)}{4!}(x-0)^4 + ...\] \[f(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + ...\]
\[f(x) = ln(1+x)\] at \(a = 0\)
\[f'(x) = \frac{1}{(1+x)} >> f'(0) = \frac{1}{(1+0)} = 1\] \[f''(x) = -\frac{1}{(1+x)^2} >> f''(0) = -\frac{1}{(1+0)^2} = -1\] \[f'''(x) = 2* \frac{1}{(1+x)^3} >> f'''(0) = 2* \frac{1}{(1+0)^3} = 2\]
\[f^{(4)}(x) = - 3 * 2 * \frac{1}{(1+x)^4} >> f^{(4)}(0) = -3* 2 * \frac{1}{(1+0)^4} = -6\]
\[f(x) = \frac{f(0)}{0!} + \frac{1(0)}{1!}(x-0) - \frac{1(0)}{2!}(x-0)^2 + \frac{2(0)}{3!}(x-0)^3 - \frac{6(0)}{4!}(x-0)^4 + ...\] \[f(x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + ...\]