Thread: taylor series

what are the first three nozero terms of the Taylor series for f(x) = cos(x) about x = -Pi?

The Taylor series expansion for a function f(x) about the point x = a is given by:

$\displaystyle f(x)=f(a)+(x-a)\frac{\partial}{\partial x} f(a) + \frac{(x-a)^{2}}{2!} \frac{\partial^{2}}{\partial x^{2}} f(a) + ...$

So in your case,
$\displaystyle f(x) = cos(x)$
and
$\displaystyle a = -\pi$

The process is just a matter of starting with the first term in the above series, inserting your function evaluated at $\displaystyle \pi$, and doing any necessary derivatives, etc, to see if it vanishes or not. Then proceed to the next term, and so on, until you have three nonvanishing terms.

Originally Posted by mr_motivator

what are the first three nozero terms of the Taylor series for f(x) = cos(x) about x = -Pi?

$\displaystyle f(x) = f(-\pi) + f'(-\pi)[x - (-\pi)] + \frac{f''(-\pi)[x - (-\pi)]^2}{2!} + \frac{f'''(-\pi)[x - (-\pi)]^3}{3!} + ...
$

$\displaystyle -1 + \frac{(x+\pi)^2}{2!} - \frac{(x+\pi)^4}{4!}$