本题目来源于试卷: Maclaurin Series，类别为 IB数学
[问答题]
Consider the function4;f5:mvdukg05vcf j jfow.7(7o/uj6 nb;u2 wfb v lrx $ f(x)=\cos (p \arccos x)$,$-1\lt x\lt 1 $ and$ p \in \mathbb{R} $.
1. Show that $ f^{\prime}(0)=p \sin \left(\frac{p \pi}{2}\right)$.
The function f and its derivative satisfy
$\left(1-x^{2}\right) f^{(n+2)}(x)=(2 n+1) x f^{(n+1)}(x)+\left(n^{2}-p^{2}\right) f^{(n)}(x), \quad n \in \mathbb{N}$
where f^{(n)}(x) denotes the n th derivative of $f(x)$ and $ f^{(0)}(x) $ is f(x) .
2. Show that $f^{(n+2)}(0)=\left(n^{2}-p^{2}\right) f^{(n)}(0)$ .
3. For $ p \in \mathbb{R} \backslash\{ \pm 1, \pm 2, \pm 3\}$ , show that the Maclaurin series for f(x) , up to and including the x^{4} term, is
$\begin{aligned}
\cos \left(\frac{p \pi}{2}\right)+p & \sin \left(\frac{p \pi}{2}\right) x-\frac{p^{2} \cos \left(\frac{p \pi}{2}\right)}{2} x^{2} \\
+ & \frac{\left(1-p^{2}\right) p \sin \left(\frac{p \pi}{2}\right)}{6} x^{3}-\frac{\left(4-p^{2}\right) p^{2} \cos \left(\frac{p \pi}{2}\right)}{24} x^{4}
\end{aligned}$
4. Hence or otherwise, find $\lim _{x \rightarrow 0} \frac{\cos (p \arccos (x))}{x} $ where p is an odd integer.
5. If p is an integer, prove the Maclaurin series for f(x) is a polynomial of .
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