# How to find the upper bound of the error in given Taylor polynomial?

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?

## 1 Answer

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.

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