The Taylor development of a function #f# at #a# is #sum_(i=1)^(oo)f^((n))(a)/(n!)(x-a)^n = f(a) + f'(a)(x-a) + f^((2))(a)/(2)(x-a)^2 +...#.

Keep in mind it's a power series so it doesn't necessarily converge to #f# or even converge somewhere else than at #x=a#.

We first need the derivatives of #f# if we want to try to write a real formula of its Taylor series.

After calculus and an induction proof, we can say that #AAk in NN : f^((2k))(x) = (-1)^(k+1)2kcos(x-1) + (-1)^(k)xsin(x-1)# and #f^((2k+1))(x) = (-1)^k((2k+1)sin(x-1) + xcos(x-1))#.

So after some rough and small simplification, it seems that the Taylor series of #f# is #sum_(k=1)^oo(-1)^(k)(xsin(x-1) -2kcos(x-1))/((2k!))(x-1)^(2k) + sum_(k=1)^oo(-1)^k((2k+1)sin(x-1) + xcos(x-1))/((2k+1)!)(x-1)^(2k+1)#.