Taylor series for $\sin^5 x$ at zero

Question

I'm trying to obtain the Taylor series for $f=\sin^5 x$ at the point $\alpha = 0$, by using the known Taylor series of $\sin x$, then taking its $5$th power, as such $$\Bigl(x-\frac{x^3}{3!}+\frac{x^5}{5!}+ \cdots \Bigl)^5.$$

Now all that needs to be done, is to calculate the coefficient of each term individually. This doesn't seems like a good solution, and I feel like I'm really misunderstanding the point of Taylor series at this point.

Anyway, a bigger confusion I have, is that if you were to start calculating the derivatives of $\sin^5(x)$, it would not really yield anything meaningful. For example ( the first derivative ) $$f^{(1)} = 5\sin^4x\cdot \cos x.$$

So $f^{(1)}(\alpha) = 0$ when $\alpha = 0$, and so the first term is just $\frac{0}{1!}\cdot(x-\alpha)^1=0$. The 2nd and third terms are $0$, as the corresponding derivatives are $0$ at $\alpha = 0$. This is not reflected in the actual series, which has non-zero terms in their place.

If we want to find the $n$th power of a known Taylor series, what are the practical steps involved?

Answer 1

Your method will work but is rather tedious. I'd much rather turn multiplication into addition using Chebyshev's formula: $$ \sin^5(x) =\frac{1}{16}(10 \sin(x) - 5 \sin(3 x) + \sin(5 x)); $$knowing the Maclaurin series for $\sin(x)$, you can get the others by direct substitution and then add up the coefficients.

Answer 2

An accurate answer is as follows.

1. The Faa di Bruno formula can be described in terms of partial Bell polynomials $B_{n,k}(x_1,x_2,\dotsc,x_{n-k+1})$ by \begin{equation}\label{Bruno-Bell-Polynomial} \frac{\textrm{d}^n}{\textrm{d} x^n}f\circ h(x)=\sum_{k=0}^nf^{(k)}(h(x)) B_{n,k}\bigl(h'(x),h''(x),\dotsc,h^{(n-k+1)}(x)\bigr). \end{equation}

2. The partial Bell polynomials $B_{n,k}$ satisfy \begin{multline}\label{bell-sin-eq} B_{n,k}\biggl(-\sin x,-\cos x,\sin x,\cos x,\dotsc, \cos\biggl[x+\frac{(n-k+1)\pi}{2}\biggr]\biggr)\\ =\frac{(-1)^k\cos^kx}{k!}\sum_{\ell=0}^k\binom{k}{\ell}\frac{(-1)^\ell}{(2\cos x)^\ell} \sum_{q=0}^\ell\binom{\ell}{q}(2q-\ell)^n \cos\biggl[(2q-\ell)x+\frac{n\pi}2\biggr] \end{multline} and \begin{multline}\label{bell-sin=ans} B_{n,k}\biggl(\cos x,-\sin x,-\cos x,\sin x,\dotsc, \sin\biggl[x+\frac{(n-k+1)\pi}{2}\biggr]\biggr)\\ =\frac{(-1)^k\sin^{k}x}{k!}\sum_{\ell=0}^k\binom{k}{\ell}\frac1{(2\sin x)^{\ell}} \sum_{q=0}^\ell(-1)^q\binom{\ell}{q}(2q-\ell)^n \cos\biggl[(2q-\ell)x+\frac{(n-\ell)\pi}2\biggr]. \end{multline} Taking $x\to0$ leads to \begin{multline}\label{bell-sin-eq=0} B_{n,k}\biggl(0,-1,0,1,\dotsc, \cos\frac{(n-k+1)\pi}{2}\biggr)\\ =\frac{(-1)^k}{k!}\biggl(\cos\frac{n\pi}2\biggr) \sum_{\ell=0}^k\frac{(-1)^\ell}{2^\ell}\binom{k}{\ell} \sum_{q=0}^\ell\binom{\ell}{q}(2q-\ell)^n \end{multline} and \begin{multline}\label{bell-sin=ans=0} B_{n,k}\biggl(1,0,-1,0,\dotsc, \sin\frac{(n-k+1)\pi}{2}\biggr)\\ =\frac{(-1)^k}{k!2^k} \biggl[\cos\frac{(n-k)\pi}2\biggr] \sum_{q=0}^k(-1)^q\binom{k}{q}(2q-k)^n =\biggl[\cos\frac{(n-k)\pi}2\biggr]2^{n-k}S_{-k/2}(n,k), \end{multline} where \begin{equation*}%\label{S(n,k,x)-satisfy-eq} S_r(n,k)=\frac1{k!}\sum_{j=0}^k(-1)^{k-j}\binom{k}{j}(r+j)^n, \quad n\ge k\ge0. \end{equation*}

3. These formulas can be applied to establish general formulas of the $n$th derivatives for functions of the types $f(\sin x)$ and $f(\cos x)$, such as $\sin^\alpha x$, $\cos^\alpha x$, $\sec^\alpha x$, $\csc^\alpha x$, $e^{\pm\sin x}$, $e^{\pm\cos x}$, $\ln\cos x$, $\ln\sin x$, $\ln\sec x$, $\ln\csc x$, $\sin\sin x$, $\cos\sin x$, $\sin\cos x$, $\cos\cos x$, $\tan x$, and $\cot x$.

4. By virtue of some formulas above-mentioned, we have \begin{align} \frac{\textrm{d}^n\sin^5x}{\textrm{d} x^n} &=\sum_{k=0}^n\langle5\rangle_k \sin^{5-k}x B_{n,k}(\cos x, -\sin x, -\cos x,\sin x,\dotsc)\\ &\to5!B_{n,5}(1, 0, -1,0,\dotsc), \quad x\to0\\ &=5! \biggl[\cos\frac{(n-5)\pi}2\biggr]2^{n-5}S_{-5/2}(n,5). \end{align} Consequently, we find \begin{equation} \sin^5x=5!\sum_{n=0}^\infty\biggl[\cos\frac{(n-5)\pi}2\biggr]\frac{2^{n-5}S_{-5/2}(n,5)}{n!}x^n =5!\sum_{k=0}^\infty(-1)^k\frac{2^{2k}S_{-5/2}(2k+5,5)}{(2k+5)!}x^{2k+5}. \end{equation}

References

1. B.-N. Guo, D. Lim, and F. Qi, Maclaurin's series expansions for positive integer powers of inverse (hyperbolic) sine and tangent functions, closed-form formula of specific partial Bell polynomials, and series representation of generalized logsine function, Appl. Anal. Discrete Math. 16 (2022), in press; available online at https://doi.org/10.2298/AADM210401017G.
2. F. Qi, Derivatives of tangent function and tangent numbers, Appl. Math. Comput. 268 (2015), 844--858; available online at https://doi.org/10.1016/j.amc.2015.06.123.
3. F. Qi and J. Gelinas, Revisiting Bouvier's paper on tangent numbers, Adv. Appl. Math. Sci. 16 (2017), no. 8, 275--281.
4. F. Qi and B.-N. Guo, An explicit formula for derivative polynomials of the tangent function, Acta Univ. Sapientiae Math. 9 (2017), no. 2, 348--359; available online at https://doi.org/10.1515/ausm-2017-0026.
5. F. Qi and B.-N. Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterr. J. Math. 14 (2017), no. 3, Art. 140, 14 pages; available online at https://doi.org/10.1007/s00009-017-0939-1.
6. Feng Qi, Da-Wei Niu, Dongkyu Lim, and Yong-Hong Yao, Special values of the Bell polynomials of the second kind for some sequences and functions, Journal of Mathematical Analysis and Applications 491 (2020), no. 2, Paper No. 124382, 31 pages; available online at https://doi.org/10.1016/j.jmaa.2020.124382.
7. F. Qi, G.-S. Wu, and B.-N. Guo, An alternative proof of a closed formula for central factorial numbers of the second kind, Turkish J. Anal. Number Theory 7 (2019), no. 2, 56--58; available online at https://doi.org/10.12691/tjant-7-2-5.
8. C.-F. Wei and F. Qi, Several closed expressions for the Euler numbers, J. Inequal. Appl. 2015, Paper No. 219, 8 pages; available online at https://doi.org/10.1186/s13660-015-0738-9.
9. A.-M. Xu and G.-D. Cen, Closed formulas for computing higher-order derivatives of functions involving exponential functions, Appl. Math. Comput. 270 (2015), 136--141; available online at https://doi.org/10.1016/j.amc.2015.08.051.
10. J.-L. Zhao, Q.-M. Luo, B.-N. Guo, and F. Qi, Remarks on inequalities for the tangent function, Hacet. J. Math. Stat. 41 (2012), no. 4, 499--506.

Answer 3

For $m,k\in\mathbb{N}=\{1,2,\dotsc\}$, we have \begin{equation}\label{sin-ell-d} \frac{\textrm{d}^m\sin^kx}{\textrm{d} x^m} =\frac{(-1)^k}{2^k} \sum_{q=0}^k(-1)^q\binom{k}{q}(2q-k)^m \cos\biggl[(m-k)\frac\pi2+(2q-k)x\biggr]. \end{equation} For $\ell\in\mathbb{N}$, we have \begin{equation}\label{sin-poer-exp} \sin^\ell x=\frac{(-1)^\ell}{2^\ell}\sum_{q=0}^\ell(-1)^q\binom{\ell}{q} \cos\biggl[(2q-\ell)x-\frac{\ell}2\pi\biggr]. \end{equation} Employing anyone of these two formulas, one can derive Maclarurin's series expansion of the function $\sin^5x$ and, generally, the function $\sin^nx$ for $n\in\mathbb{N}$.

These two formulas can be found in the following paper:

1. Bai-Ni Guo and Feng Qi, On the Wallis formula, International Journal of Analysis and Applications 8 (2015), no. 1, 30--38.

Answer 4

A general answer to this question is the following result.

Theorem. For $\ell\in\mathbb{N}$ and $x\in\mathbb{R}$, we have \begin{equation}\label{sine-power-ser-expan-eq} \biggl(\frac{\sin x}{x}\biggr)^{\ell} =1+\sum_{j=1}^{\infty} (-1)^{j}\frac{T(\ell+2j,\ell)}{\binom{\ell+2j}{\ell}} \frac{(2x)^{2j}}{(2j)!}, \end{equation} where \begin{equation}\label{T(n-k)-EF-Eq(3.1)} T(n,k)=\frac1{k!} \sum_{\ell=0}^{k}(-1)^{\ell}\binom{k}{\ell}\biggl(\frac{k}2-\ell\biggr)^n. \end{equation} denotes central factorial numbers of the second kind.

For details of the proof for the above result, please read the arXiv preprint below.

Reference

1. Feng Qi and Peter Taylor, Several series expansions for real powers and several formulas for partial Bell polynomials of sinc and sinhc functions in terms of central factorial and Stirling numbers of second kind, arXiv preprint (2022), available online at https://arxiv.org/abs/2204.05612v4 or https://doi.org/10.48550/arXiv.2204.05612.