# Question 059f6

Dec 22, 2015

f(x) = 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)

#### Explanation:

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 $\forall k \in \mathbb{N} : {f}^{\left(2 k\right)} \left(x\right) = {\left(- 1\right)}^{k + 1} 2 k \cos \left(x - 1\right) + {\left(- 1\right)}^{k} x \sin \left(x - 1\right)$ and ${f}^{\left(2 k + 1\right)} \left(x\right) = {\left(- 1\right)}^{k} \left(\left(2 k + 1\right) \sin \left(x - 1\right) + x \cos \left(x - 1\right)\right)$.

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)#.