Prove there exists a constant $K>0$ such that $|e^z-1-z-\frac{z^2}{2}|<K|z^3|$ as $z \to 0$

Question

The title says it all however:

Prove that there exists a positive constant $K$ such that $|e^z-1-z-\frac{z^2}{2}|<K|z^3|$ when $|z|$ is sufficiently small.

Or in other words prove $e^x=1+x+\frac{x^2}{2}+O(x^3)$ by the way I have no background in asymptotics so complicated solutions will have no benefit for me.

Answer 1

We have

$$e^z - 1 - z - \frac{z^2}{2} = \sum_{n = 3}^\infty \frac{z^n}{n!},$$

hence

$$\biggl\lvert e^z - 1 - z - \frac{z^2}{2}\biggr\rvert \leqslant \sum_{n = 3}^\infty \frac{\lvert z\rvert^n}{n!}.$$

If we require $\lvert z\rvert < 1$, then we have $\lvert z\rvert^n < \lvert z\rvert^3$ for all $n \geqslant 3$, and that gives you an explicit $K$.