# Forget about the $\cos,\sin$ function, show that $\left|1-x^2/2!+x^4/4!-x^6/6!+…\right|\leq1$

Forget about the $\cos,\sin$ function, show that $\left|1-x^2/2!+x^4/4!-x^6/6!+...\right|\leq1$

-
Why doesn't differentiation help? –  Ryan Sep 1 '13 at 2:48

Define $$f(x)=\sum_{n\geqslant 0} (-1)^n\frac{x^{2n}}{(2n)!}$$
One can readily see this converges for any $x$. Note that $$f''+f=0$$ whence $$f'f''+ff'=0$$ which means $$f'^2+f^2=K$$
By plugging in values we see $K=1$; thus $$f'^2+f^2=1$$ whence we must have $|f'|,|f|\leq 1$ for all $x$.
The weird thing, to me, is that the first several terms can actually get large for $|x|\gg1$, but then things have to cancel out somehow. –  dfeuer Sep 1 '13 at 3:00