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If Taylor polynomial for $\sin(x)$ is $\sum_{n=0}^{+\infty} \frac{(-1)^{n}}{(2n+1)!}x^{2n+1} $. What do I have to do to find what degree of Taylor polynomial I have to use so the error is not greater than $10^{-4}$ in approximation of $\sin(100)$?

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  • $\begingroup$ It's true only around 0. $\endgroup$ – Archis Welankar Jan 26 '20 at 15:46
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    $\begingroup$ Well, since the series alternates, and decreases for sufficiently large terms, you could just use Sterling's approximation. But surely it is more efficient to use periodicity...$32\pi\approx 100.5309649$ helps. $\endgroup$ – lulu Jan 26 '20 at 15:48
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    $\begingroup$ Why using this series in practice is a really bad idea: the largest term in the alternating series is about $\approx 100^{100}/100! \approx 10^{42}$. Thus if you are to use the alternating series to compute it you would need to add about $300$ numbers that would be up to $47$ digits long (you also need $5$ decimal digits to get the desired precision). This would not even work on a computer unless you used arbitrary precision numbers. $\endgroup$ – Winther Jan 26 '20 at 16:03
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    $\begingroup$ Your remainder term is not correct. It needs a factor $x^n$ in it, which is huge and why Winther gets that you need so many terms $\endgroup$ – Ross Millikan Jan 26 '20 at 17:06
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    $\begingroup$ Without using the value of $\pi$, you can use angle reduction using the double-angle trigonometric identities. Set $x=100/2^6=1.5625$, then compute $\sin x$ and $\cos x$ with an error less than $10^{-8}$ using the Taylor expansions, and then reconstruct the wanted quantities using $\sin(2^{k+1}x)=2\sin(2^kx)\cos(2^kx)$ and $\cos(2^{k+1}x)=\cos^2(2^kx)-\sin^2(2^kx)$ or similar. $\endgroup$ – Lutz Lehmann Jan 26 '20 at 19:29
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It is a(n eventually) converging alternating series, so the alternating series theorem applies. Find the first term after it starts decreasing that is less than $10^{-4}$ in magnitude and you are done.

Probably you are expected to look up the error term for the Taylor series. Note that all the derivatives of $\sin x$ are less than $1$ in magnitude, so you can ignore that.

As the comments point out, you will get there with many fewer terms if you are allowed to center the Taylor series at $32\pi$ instead of $0$.

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  • $\begingroup$ If it's not mentioned where the Taylor series is centered at, what do I use as a default for this particular problem? $\endgroup$ – Exzone Jan 26 '20 at 16:02
  • $\begingroup$ The series you quote is centered at $0$. The closer you can center the series to the point of interest the better because all those $x^i$ terms in your series become $(x-c)^i$ where $c$ is the point you center at. As you can easily calculate trig functions of $32\pi$ that is a good place. $\endgroup$ – Ross Millikan Jan 26 '20 at 16:04

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