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We used this polynomial $P_2(x)=1-\frac{1}{2}*x^2$ to approximate the function $f(x)=\cos x$ in the interval $[\frac{-1}{2},\frac{1}{2}]$

How do I find the upper bound of the error?

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The cosine series is alternating. This means that the approximation error is bounded by the next term in the series, $\dfrac{x^4}{24}$.

This is a consequence of the Leibniz test for the convergence of alternating series. You need to show that the sequence $\dfrac{x^{2k}}{(2k)!}$, $k\ge 1$, is falling, the claim is then a consequence.

The same argument about the partial sums of an alternating series tells you that $$ \frac{x^4}{24}-\frac{x^6}{720}\le \cos(x)-1+\frac{x^2}2\le \frac{x^4}{24} $$ so that this error bound can not be much improved over the given interval.

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  • $\begingroup$ thanks for the solution, but the problem is that I'm having problem how to come to that solution. $\endgroup$ – Betim Shala Dec 12 '18 at 9:36
  • $\begingroup$ I hope that you know about the convergence tests for series? $\endgroup$ – LutzL Dec 12 '18 at 9:45

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