General Formula for a Taylor Series Expansion

\[\begin{align*}f\left( x \right) & = \sum\limits_{n = 0}^\infty {\frac{{{f^{\left( n \right)}}\left( a \right)}}{{n!}}{{\left( {x - a} \right)}^n}} \\ & = f\left( a \right) + f'\left( a \right)\left( {x - a} \right) + \frac{{f''\left( a \right)}}{{2!}}{\left( {x - a} \right)^2} + \frac{{f'''\left( a \right)}}{{3!}}{\left( {x - a} \right)^3} + \cdots \end{align*}\]

Question One

\[f(x) = \frac{1}{(1-x)}\]

\[f(a)\quad =\quad \frac { 1 }{ 1\quad -\quad a }\quad\quad f(0) = 1\] \[{ f }^{ \prime }(a)\quad =\quad \frac { 1 }{ { (1-a) }^{ 2 } }\quad\quad f^{(1)}(0) = 1\] \[{ f }^{ \prime \prime }(a)\quad =\quad \frac { 2 }{ { (1-a) }^{ 3 } }\quad\quad f^{(2)}(0) = 2\] \[{ f }^{ \prime \prime \prime}(a)\quad =\quad \frac { 6 }{ { (1-a) }^{ 4 } }\quad\quad f^{(3)}(0) = 6\] \[{ f }^{(4)}(a)\quad =\quad \frac { 24 }{ { (1-a) }^{ 5 } }\quad\quad f^{(4)}(0) = 24\]

Question Two

\[f(x) = e^x\]

\[f'(x) = e^x; f''(x) = e^x; f'''(x) = e^x\] \[f^{(n)}(x) = e^x\] \[f^{(n)}(0) = 1\] \[e^x = \sum \frac{f^{(n)}(0)}{n!}x^n\] \[= \sum \frac{1}{n!}x^n\]

Question Three

\[f(x) = ln(1 + x)\]

\[f'(x) = \frac{1}{(x + 1)}\] \[f''(x) = \frac{-1}{(x + 1)^2}\] \[f'''(x) = \frac{2}{(x + 1)^3}\] \[f''''(x) = \frac{-6}{(x + 1)^4}\] \[f^{(n)}(x) = \frac{(-1)^{n-1}(n-1)!}{(x + 1)^n}\] \[f^{(n)}(0) = (-1)^{n-1}(n-1)!\] \[ln(x + 1) = \sum \frac{f^{(n)}(0)}{n!}x^n\] \[= \sum \frac{(-1)^{n-1}(n-1)!}{n!}x^n\] \[= x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} ...\]