2022-06-28

What is the Taylor series of
$\mathrm{sin}\left(5z\right)-5\mathrm{sin}\left(z\right)$
at ${z}_{0}=\pi$
I am confused whether to substitute $5\left(z-\pi \right)$ or $5z-\pi$ into the Maclaurin series of $\mathrm{sin}\left(z\right)$

Expert

Substitute $5\left(z-\pi \right)$ and observe that
$\mathrm{sin}\left(5z\right)=-\mathrm{sin}\left(5\left(z-\pi \right)\right)$ and $\mathrm{sin}\left(z\right)=-\mathrm{sin}\left(z-\pi \right)$

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