Shift a power series to be centered at a

power seriessequences-and-seriestaylor expansion

Let's say that I want to approximate $\sin{(4)}$ Using a the first three non-zero terms of a power series. If I'm centered at $x=0$, I get a bad approximation, as you can see in the image.

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How do I get a better approximation, not by adding more terms to the power series, but by finding a different power series centered at $x=4$? If I just use the Taylor Series, the first term would be $\sin{(4)}$, but that defeats the purpose – I can't use $\sin{(4)}$ if I'm trying to approximate $\sin{(4)}$. Thank you!

Best Answer

Per the suggestions in the comments, I have expanded the series around $\pi$. Including terms up through $x^3$ seems to approximate $\sin 4$ pretty well already.

The even-numbered terms are multiples of $\sin \pi = 0$, so I have omitted them from this display.

graph of first three nonzero terms of Taylor series for sin(x) around x=\pi graph of sin(x), sum of terms up to x^3, sum of terms up to x^5

Interactive hilarity

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