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Let $f(x) = \sin(x)$ and $g(x) = \cos(x^3)$.

Let $h(x) = \sin(x) + \cos(x^3)$.

The function $h$ can be represented as the power series:

$$h(x)=\sum_{n=0}^\infty a_nx^n$$

which of course is its Taylor series centered at $a = 0$. Identify $a_n$.

I was advised against computing derivatives of $h$, and told I would find working through this with piecewise functions more doable instead. However, I'm completely lost as to where to even start with this question. I do know the basics of Taylor series and how to derive a series from, say, $\sin(x)$, but I'm stumped with this.

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So, you know that$$\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}-\cdots$$and I suppose that you know that$$\cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots$$Now note the it follows from this last equality that$$\cos(x^3)=1-\frac{x^6}{2!}+\frac{x^{12}}{4!}-\frac{x^{12}}{6!}+\cdots$$Can you take it from here?

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