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I attempted to solve this problem:

Determine the values of $x$ for which the function can be replaced by the Taylor polynomial if the error cannot exceed $0.001$.

$$f(x)=\cos(x)\approx 1-\frac{x^2}{2!}+\frac{x^4}{4!}$$

I thought this equation went to $n=4$, so when using Lagrange Error, I found the max of the fifth derivative ($n+1=4+1=5$). However, I found out it was wrong; actually, this equation went to $n=5$ and I should find the max of the sixth derivative for Lagrange Error. Why is that? It seems not to follow the $n+1$ rule for Lagrange Error.

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It's valid to use an error term based on the fifth derivative here (and get some range of $x$ values that way).

However, you could also notice that the coefficient of $x^5$ in the Taylor expansion of $\cos(x)$ is $0$, and so you can also think of this Taylor expansion as going out to the fifth term (which is $0 \cdot x^5$) and use an error term based on the sixth derivative.

Both approaches will give you true bounds on $x$-values, but we expect the second approach to produce better results (that is, identify more values of $x$ for which the approximation is good enough), because the sixth-derivative error term tends to be smaller than the fifth-derivative error term.

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