Find the Maclaurin series for the function f(x)=sin5x. Use the table of power series for elementary functions

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Series
Find the Maclaurin series for the function $$f(x)=\sin5x$$. Use the table of power series for elementary functions

2021-02-15
Given:
$$f(x)=\sin5x$$
From the table of power series, the Maclaurin series for the above function using the table of power series.
$$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+...=\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)!}$$
Replacing x by 5x,
$$\sin5x=5x-\frac{(5x)^3}{3!}+\frac{(5x)^5}{5!}+\frac{(5x)^7}{7!}+...$$
$$\sin5x=5x-\frac{125x^3}{6}+\frac{325x^5}{24}+\frac{15625x^7}{1008}+...$$
$$\sin5x=\sum_{n=0}^\infty\frac{(-1)^n(5x)^{2n+1}}{(2n+1)!}$$
Hence, $$\sin5x=\sum_{n=0}^\infty\frac{(-1)^n(5x)^{2n+1}}{(2n+1)!}$$

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