Use the Taylor series for $\sin(x)$ to find a Taylor series for $f(x) = p(\sin x) $ around point $x=0$, where $p(x) = 4x^3 - 3x$

Question

A question asks:

Consider the polynomial $p(x) = 4x^3 - 3x.$ Use the Taylor series for $\sin(x)$ to find a Taylor series for $f(x) = p(\sin x) $ around point $x=0$.
Don't just calculate this by differentiating right away. Instead, start with the first few terms of the Taylor series for $\sin(x)$, and substitute those into the given polynomial to figure out the first few terms.

I am not sure what exactly they want me to do here. The first few terms of $\sin x$ are: $$ x - \frac{x^3}{6} + \frac{x^5}{120} -+\cdots.$$

Are they just looking for me to plug this expression into the original polynomial?

The answer that the site linked below shows is $$-3x + \frac{9}{2}x^3+ \frac{-81}{40}\,x^5 . . .$$

Were they actually cubing trinomials to find the answer, or am I missing an easier method?

https://ximera.osu.edu/course/kisonecat/multivariable-calculus/master/taylor-series/remainders

Answer 1

Hint:

First show that

$$4\sin^3(x)-3\sin(x)=-\sin(3x)$$

Then show that

$$f(x)=-\sin(3x)=\sum_{n=0}^\infty\frac{(-1)^{n+1}(3x)^{2n+1}}{(2n+1)!}$$

Answer 2

It is known that $$ \sin x=\sum_{k=0}^\infty(-1)^k\frac{x^{2k+1}}{(2k+1)!}, \quad x\in\mathbb{R}. $$ On page 43 in the handbook [1] below, there is the series expansion $$ \sin^3x=\frac14\sum_{k=1}^\infty(-1)^{k+1}\bigl(3^{2k+1}-3\bigr)\frac{x^{2k+1}}{(2k+1)!}, \quad x\in\mathbb{R}. $$ Consequently, we obtain $$ p(\sin x)=4\sin^3x-3\sin x=\sum_{k=1}^\infty(-1)^{k+1}\frac{(3x)^{2k+1}}{(2k+1)!}, \quad x\in\mathbb{R}. $$

Reference

1. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Translated from the Russian, Translation edited and with a preface by Daniel Zwillinger and Victor Moll, Eighth edition, Revised from the seventh edition, Elsevier/Academic Press, Amsterdam, 2015; available online at https://doi.org/10.1016/B978-0-12-384933-5.00013-8.

Answer 3

A general and alternative answer to this kind of questions is the following theorem obtained in Lemma 2 of the paper [1] below.

For $\ell\in\mathbb{N}$ and $x\in\mathbb{R}$, assume that the value of the function $\bigl(\frac{\sin x}{x}\bigr)^{\ell}$ at the point $x=0$ is $1$. Then \begin{equation}\label{sine-power-ser-expan-eq} \biggl(\frac{\sin x}{x}\biggr)^{\ell} =\sum_{j=0}^{\infty} (-1)^{j}\frac{R\bigl(\ell+2j,\ell,-\frac{\ell}{2}\bigr)}{\binom{\ell+2j}{\ell}} \frac{(2x)^{2j}}{(2j)!}, \end{equation} where $R\bigl(\ell+2j,\ell,-\frac{\ell}{2}\bigr)$ is given by\begin{equation}\label{S(n,k,x)-satisfy-eq} R(n,k,r)=\frac1{k!}\sum_{j=0}^k(-1)^{k-j}\binom{k}{j}(r+j)^n \end{equation} for $r\in\mathbb{R}$ and $n\ge k\ge0$.

The notation $R(n,k,r)$ stands for weighted Stirling numbers of the second kind, which was first introduced in [2] below.

References

1. F. Qi, Series expansions for any real powers of (hyperbolic) sine functions in terms of weighted Stirling numbers of the second kind, arXiv (2022), available online at https://arxiv.org/abs/2204.05612 or https://doi.org/10.48550/arXiv.2204.05612.
2. L. Carlitz, Weighted Stirling numbers of the first and second kind, I, Fibonacci Quart. 18 (1980), no. 2, 147--162.