Is $\cos(x) \geq 1 - \frac{x^2}{2}$ for all $x$ in $R$? I encountered this question:
Is $\cos(x) \geq 1 - \frac{x^2}{2}$   for all $x$ in  $R$  ?
I draw to myself the function graphs but i can't find the right way to answer this type of questions.
Cosx function graph
1-x^2/2 function graph

Consider the definition of $\cos x$ as the sum of a power series:
$$\cos x=\sum_{k\ge0}(-1)^k\frac{x^{2k}}{(2k)!}.$$
As $\cos x$ is an even function, and the series has terms of even degree only, we may suppose $x\ge 0$.
Furthermore, we may suppose  $x\le 2$ because $1-\frac{x^2}2<-1$ for $x>2$. For each  $x\in[0,2]$, it's an alternating series with  terms decreasing in magnitude. Partial sums are approximations of $\cos x$ with an error
$$\cos x-\biggl(\sum_{k=0}^n(-1)^k\frac{x^{2k}}{(2k)!}\biggr)= (-1)^{n+1}\frac{x^{2(n+1)}}{(2(n+1))!}+\dotsm$$
which has the sign of the first omitted term (and its absolute value is not more than the absolute value of this term).
Hence $\;1-\dfrac{x^2}{2}<\cos x$.
