<s>[INST] Proof about exp function, inequality I need to proof the inequality, that $e^x > 1+x^2$ for positive $x$?
Thanks
Taylor doesn't work
Easy to proof for  (0,1]. So one can assume x>1 in his proof
 [/INST] Consider the Taylor expansion
$$ e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \cdots. $$
If $x \leq 2$ then $x \geq x^2/2$ and so $e^x > 1 + x + x^2/2 \geq 1 + x^2/2 + x^2/2 = 1 + x^2$.
If $x > 2$ then $x^3/6 > x^2/3$ and $x^4/24 > x^2/6$ and so $e^x > 1 + x^2/2 + x^3/6 + x^4/24 > 1 + x^2/2 + x^2/3 + x^4/6 = 1 + x^2$.
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