Why is the Taylor expansion of $\cos$ decreasing ?
$\cos(t)=1-\frac{t^2}{2!}+\frac{t^4}{4!}-\frac{t^6}{6!}+...$
such that one can estimate $\cos(t)<1-\frac{t^2}{2!}+\frac{t^4}{4!}$
I saw that a similar estimation is valid for $\sin(t)$ $(\sin(t)>t-\frac{t^3}{3!}$)
For the cosinus case :If we choose $t$ very big, s.t. ($-\frac{t^6}{6!}+\frac{t^8}{8!}>0)$, in general for finitely many such pairs, why must the rest always prevail ?
Is there a rigorous proof, without landau symbols ?
The result is clear if $0\lt t\le 1$ ("alternating series"). However, it is true for all $t$. For simplicity, in what follows we will take $t\ge 0$. The argument is an interesting instance of lifting ourselves up by our own bootstraps.
Let $t$ be positive. First we show that $\sin t \lt t$. Let $f_1(t) =t-\sin t$. We have $f_1(0)=0$ and $f_1'(t)=1-\cos t \ge 0$ for all positive $t$. In fact $f_1'(t)\gt 0$ except at the points $2n\pi$, so $f_1(t)$ is strictly increasing, and therefore $\sin t\lt t$ for $t\gt 0$.
Next we show that $\cos t\gt 1-\frac{t^2}{2!}$ for $t\gt 0$. Let $g_1(t)=\cos t -\left(1-\frac{t^2}{2!}\right)$. Then $g_1(0)=0$, and $g_1'(t)=-\sin t +t \gt 0$ for $t\gt 0$. It follows that $g_1(t)$ is increasing, and therefore $\cos t\gt 1-\frac{t^2}{2!}$ for all $t\gt 0$.
Next we show that $\sin t\gt t-\frac{t^3}{3!}$ for all positive $t$. Let $f_2(t)=\sin t-\left(t-\frac{t^3}{3!}\right)$. We have $f_2(0)=0$. Also, $f_2'(t)=\cos t -\left(1-\frac{t^2}{2!}\right)\gt 0$. Thus $f_2(t)$ is increasing for $t\gt 0$, and therefore $\sin t\gt t-\frac{t^3}{3!}$ for all positive $t$.
Next we show that $\cos t\lt 1-\frac{t^2}{2!}+\frac{t^4}{4!}$ for all $t\gt 0$. Let $g_2(t)=\left(1-\frac{t^2}{2!}+\frac{t^4}{4!}\right)-\cos t$. Then $g_2(0)=0$. Also, $g_2'(t)= \sin t-\left( t-\frac{t^3}{3!}\right)\gt 0$ for all positive $t$. So $g_2(t)$ is increasing, and the desired result follows.
Remark: we can also get the result by using Taylor's Theorem. The above calculations were an exercise in writing a proof that uses only tools available early on in a calculus course.