I'd like to prove the following via Taylor approximation :
$$x > 0 \implies \sin x > x-\frac{x^3}{3!}$$
I tried to estimate the error but I found $E_3(x) \leq |\frac{x^4}{4!}| $, which doesn't give information about the sign of the error. Then, I tried to compare the sign of each couple of terms in the error but it gives the right result just when $0<x \leq1 $ because in this case
$$(\frac{x^5}{5!} – \frac{x^7}{7!} )+(\frac{x^9}{9!} -\frac{x^{11}}{11!} ) + \dots$$
all the couples are positive. But from here I don't know how to continue to include the case $x>1$.
How can I find a way using Taylor approximation ? (I know other methods like derivatives can be used, but I want just to use Taylor if possible).
Besides, Is there a general method to find the intervals where the sign of the error is positive/negative?
Best Answer
Hint
If $x\in [0,\pi]$, there is $b\in [0,x]$ s.t. $$\sin(x)=x-\frac{x^3}{3!}+\frac{\sin(b)}{4!}x^4.$$
If $x\geq \pi$ then $x-\frac{x^3}{3!}< -1$.