ASSIGNMENT 14 - TAYLOR SERIES

This week, we’ll work out some Taylor Series expansions of popular functions.

1. \(f\left( x \right) =\frac { 1 }{ (1-x) }\)

\[f\left( x \right) =\sum { \begin{matrix} \infty \\ 0 \end{matrix} } ({ f }^{ n }(0)/n!){ x }^{ n }\] \[f(0)=1\] \[f`(0)=1\] \[f``(0)=2\] \[f```(0)=6\] \[f````(0)=24\] \[f\left( x \right) =1+{ x }^{ 1 }+{ x }^{ 2 }+{ x }^{ 3 }+{ x }^{ 4 }+...{ x }^{ n }\]

2. \(f\left( x \right) ={ e }^{ x }\)

\[f\left( x \right) =\sum { \begin{matrix} \infty \\ 0 \end{matrix} } ({ f }^{ n }(0)/n!){ x }^{ n }\] \[f(0)={ e }^{ 0 }=1\] \[f`(0)=1\] \[f``(0)=1\] \[f```(0)=1\] \[f````(0)=1\] \[f\left( x \right) =1+{ x }^{ 1 }+{ x }^{ 2 }/2!+{ x }^{ 3 }/3!+{ x }^{ 4 }/4!+...{ x }^{ n }/n!\]

3.\(f\left( x \right) ={ ln(1+x) }\)

\[f\left( x \right) =\sum { \begin{matrix} \infty \\ 0 \end{matrix} } ({ f }^{ n }(0)/n!){ x }^{ n }\] \[f\left( x \right) ={ ln(1+0)=0 }\] \[f\left( x \right) ={ 1/(1+x),f`(0)=1 }\] \[f\left( x \right) ={ -1/{ (1+x) }^{ 2 },f``(0)=-1 }\] \[f\left( x \right) ={ 2/{ (1+x) }^{ 3 },f```(0)=2 }\] \[f\left( x \right) ={ 2.3/{ (1+x) }^{ 4 },f````(0)=-6 }\] \[f\left( x \right) ={ x }^{ 1 }-{ x }^{ 2 }/2!+{ x }^{ 3 }/3!-{ x }^{ 4 }/4!+...{ x }^{ n }/n!\]