# Find the Maclaurin series for the function f(x)=2sin x^3 Use the table of power series for elementary functions

Series
Find the Maclaurin series for the function $$\displaystyle{f{{\left({x}\right)}}}={2}{{\sin{{x}}}^{{3}}}$$
Use the table of power series for elementary functions

2021-02-26
To find the Maclaurin series for the function $$\displaystyle{f{{\left({x}\right)}}}={2}{{\sin{{x}}}^{{3}}}$$
Solution:
From the given table, we can find that Maclaurin series of the function $$\displaystyle{\sin{{x}}}$$ is:
$$\displaystyle{\sin{{x}}}={x}-{\frac{{{x}^{{3}}}}{{{3}!}}}+{\frac{{{x}^{{5}}}}{{{5}!}}}-{\frac{{{x}^{{7}}}}{{{7}!}}}+\ldots$$
Now, if we replace x by $$\displaystyle{x}^{{3}}$$ then:
$$\displaystyle{{\sin{{x}}}^{{3}}=}{x}^{{3}}-{\frac{{{\left({x}^{{3}}\right)}^{{3}}}}{{{3}!}}}+{\frac{{{\left({x}^{{3}}\right)}^{{5}}}}{{{5}!}}}-{\frac{{{\left({x}^{{3}}\right)}^{{7}}}}{{{7}!}}}+\ldots$$
$$\displaystyle={x}^{{3}}-{\frac{{{x}^{{9}}}}{{{3}!}}}+{\frac{{{x}^{{{15}}}}}{{{5}!}}}-{\frac{{{x}^{{{21}}}}}{{{7}!}}}+\ldots$$
Therefore, Maclaurin series of the function $$\displaystyle{f{{\left({x}\right)}}}={2}{{\sin{{x}}}^{{3}}}$$ will be:
$$\displaystyle{2}{{\sin{{x}}}^{{3}}=}{2}{\left[{x}^{{3}}-{\frac{{{x}^{{9}}}}{{{3}!}}}+{\frac{{{x}^{{{15}}}}}{{{5}!}}}-{\frac{{{x}^{{{21}}}}}{{{7}!}}}+\ldots\right]}$$
$$\displaystyle={2}{x}^{{3}}-{\frac{{{2}{x}^{{9}}}}{{{6}}}}+{\frac{{{2}{x}^{{{15}}}}}{{{120}}}}-{\frac{{{2}{x}^{{{21}}}}}{{{5040}}}}+\ldots$$
$$\displaystyle={2}{x}^{{3}}-{\frac{{{x}^{{9}}}}{{{3}}}}+{\frac{{{x}^{{{15}}}}}{{{60}}}}-{\frac{{{x}^{{{21}}}}}{{{2520}}}}+\ldots$$
Hence, Maclaurin series of the given function is
$$\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{3}}-{\frac{{{x}^{{9}}}}{{{3}}}}+{\frac{{{x}^{{{15}}}}}{{{60}}}}-{\frac{{{x}^{{{21}}}}}{{{2520}}}}+\ldots$$