Prove that the error of the Taylor expansion of satisfies the following inequality.

*Proof.* Since the derivatives of are always , or we know that for we have . (In other words, the st derivative is bounded above by 1 and below by .) Therefore, we can apply Theorem 7.7 (p. 280 of Apostol) to estimate the error in Taylor’s formula at with and . For this gives us

Next, (from the second part of Theorem 7.7) if we have

the last inequality is false cause |E|>0 and x^{2n+1}<0

For the x<0 part, shouldn't 'x' be '-x'? The definition uses |x|.