Maclaurin’s Series

For a function f(x) for which all derivatives evaluated at x = a exist,

\(f(x)=f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f”(a)}{2!} (x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+ \cdots.\)

 

Maclaurin’s expansion
\(sin x &= \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1}\)

For a function f(x) for which all derivatives evaluated at x = a exist,

\(\sin x &= \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1} &&= x – \frac{x^3}{3!} + \frac{x^5}{5!} – \cdots\)

 then \(f(x)=f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f”(a)}{2!} (x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+ \cdots.+\frac{f^{(n)}(a)}{n!}(x-a)^n+{ R }_{ n }(x)\)

where  \({ R }_{ n }(x)=\frac { f^{ (n+1) }(c) }{ (n+1)! } (x-a)^{ n+1 }\quad for\quad some\quad c\in \left( a,x \right)\)

 

1. f(x) = Sin(x)

 

2. f(x) = Cos(x)

3. f(x) = ln(x)