McLaren F1 and Taylor Series

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Taylor Series

\begin{aligned}

&\text{Taylor Series (about }x=a\text{):}\\

&\quad f(x)

= f(a)

+ f'(a)\,(x-a)

+ \frac{f''(a)}{2!}\,(x-a)^{2}

+ \cdots

+ \frac{f^{(n)}(a)}{n!}\,(x-a)^{n}

+ \cdots

\\[8pt]

&\text{Maclaurin Series (special case }a=0\text{):}\\

&\quad f(x)

= f(0)

+ f'(0)\,x

+ \frac{f''(0)}{2!}\,x^{2}

+ \cdots

+ \frac{f^{(n)}(0)}{n!}\,x^{n}

+ \cdots

\\[6pt]

&\text{The }n\text{-th derivative at the center }a

\text{ (or }0\text{) divided by }n!\\

&\quad \text{is the coefficient of }(x-a)^{n}\,\text{which is the value of the derivative}.

\end{aligned}

\frac{1}{1 - x}

\begin{aligned}

&\frac{1}{1 - x}

= \sum_{n=0}^{\infty} x^n

= 1 + x + x^2 + \cdots + x^n + \cdots

\quad(\lvert x\rvert < 1)

\end{aligned}

f(x)=\frac{1}{1+x}

\frac{1}{1+x}

=1 - x + x^2 - x^3 + \cdots

=\sum_{n=0}^{\infty}(-1)^n x^n,

\quad |x|<1.

a_n = (-1)^n x^n,\quad n=0,1,2,\dots

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

\ln(1+x)

= x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots

=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{x^n}{n},

\quad |x|<1.

a_n = (-1)^{n-1}\frac{x^n}{n},\quad n=1,2,3,\dots

f(x)=\arctan x

\arctan x

= x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots

=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1},

\quad |x|\le1.

a_n = (-1)^n\frac{x^{2n+1}}{2n+1},\quad n=0,1,2,\dots

f(x)=e^x

e^x

=1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots

=\sum_{n=0}^{\infty}\frac{x^n}{n!},

\quad x\in\mathbb{R}.

a_n = \frac{x^n}{n!},\quad n=0,1,2,\dots

f(x)=\sin x

\sin x

= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots

=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!},

\quad x\in\mathbb{R}.

a_n = (-1)^n\frac{x^{2n+1}}{(2n+1)!},\quad n=0,1,2,\dots

f(x)=\cos x

\cos x

=1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots

=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!},

\quad x\in\mathbb{R}.

a_n = (-1)^n\frac{x^{2n}}{(2n)!},\quad n=0,1,2,\dots