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

I tried to use differentiation, but it doesn't seems helpful. Please help. Thanks.

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

1 Answer 1

up vote 12 down vote accepted

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 $x$.

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Readily because it's alternating and eventually strictly decreasing in absolute value? –  dfeuer Sep 1 '13 at 2:54
    
@dfeuer It is absolutely convergent. –  Pedro Tamaroff Sep 1 '13 at 2:55
    
You could use the ratio test to prove absolute convergence or just quickly use the AST for conditional convergence. –  Jon Claus Sep 1 '13 at 2:55
    
I just realized the ratio test does it. –  dfeuer Sep 1 '13 at 2:57
    
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

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