When centered at \(c=\pi/2\)
\[\begin{equation} f'(0) = -sin(x) =-1 \end{equation}\] \[\begin{equation} f''(0) = -cos (x) =0 \end{equation}\] \[\begin{equation} f'''(0) = sin(x) =1 \end{equation}\] \[\begin{equation} \frac{d^4}{d(x)^4} sin(x) = cos (x) =0 \end{equation}\] \[\begin{equation} \frac{d^5}{d(x)^5} cos x = -sin(x) =-1 \end{equation}\]…
\[\begin{equation} \begin{split} \displaystyle\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n &= cos (\pi/2) +\frac{-sin(\pi/2)}{1!}(x-\pi/2)^1+\frac{-cos(\pi/2)}{2!}(x-\pi/2)^2+\frac{sin(\pi/2)}{3!}(x-\pi/2)^3+\frac{cos(\pi/2)}{4!}(x-\pi/2)^4...\\ &= 0+\frac{-1}{1!}(x-\pi/2)^1+\frac{1}{3!}(x-\pi/2)^3+\frac{-1}{5!}(x-\pi/2)^5...\\ &= -(x-\pi/2)^1+\frac{1}{3!}(x-\pi/2)^3-\frac{1}{5!}(x-\pi/2)^5...\\ &= \displaystyle\sum_{n=0}^{\infty}(-1)^{n+1}(-1)^(n+1)\frac{1}{(2n+1)!}x^n \end{split} \end{equation}\] \[\begin{equation} \begin{split} \end{split} \end{equation}\]library(pracma)
f <- function(x) cos(x)
p <- taylor(f, pi/2, 4)
p ## [1] -9.855883e-10 1.666666e-01 -7.853981e-01 2.337004e-01 9.248323e-01