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

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

Question
Series
asked 2021-02-14
Find the Maclaurin series for the function \(f(x)=\sin5x\). Use the table of power series for elementary functions

Answers (1)

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, \(\sin5x=\sum_{n=0}^\infty\frac{(-1)^n(5x)^{2n+1}}{(2n+1)!}\)
0

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