$ f(x) = 6x^7\sin^2(x^{1000}) e^{x^2} $
Find $ f^{(2013)}(0) $
A math forum friend suggest me to use big O symbol, however have no idea what that is, so how does that helping?
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11$\begingroup$ Where did you find this problem? 2013 is an odd year to choose... :) $\endgroup$ – apnorton Dec 30 '12 at 20:58
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6$\begingroup$ $\sin^2(x^{1000}) = x^{2000}+O(x^{4000})$ reduces the complexity of the computation considerably... $\endgroup$ – WimC Dec 30 '12 at 21:04
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Note that,
$$ 6\,x^{7} \sin\left(x^{1000}\right)\sin\left(x^{1000}\right)e^{x^2} $$
$$ = 6\,x^{7} \left( x^{1000}-\frac{x^{3000}}{3!}+\dots \right)\left( x^{1000}-\frac{x^{3000}}{3!}+\dots \right)\left(1+\frac{x^2}{1!}+\frac{x^4}{4!}+\dots\right) $$
$$ = 6x^7x^{2000}\left( 1-\frac{x^{2000}}{3!} +\dots\right)^2\left(1+\frac{x^2}{1!}+\frac{x^4}{2!}+\dots\right) $$
$$ = 6x^{2007}\left(1+\frac{x^2}{1!}+\frac{x^4}{2!}+\frac{x^6}{3!}+\dots\right)\left( 1-\frac{x^{2000}}{3!} +\dots\right)^2 $$
Now, it is clear that the coefficient of $x^{2013}$ is $1$, which implies that
$$ \frac{f^{(2013)}(0)}{(2013)!} = 1 \implies f^{(2013)}(0)=(2013)!. $$
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Hint:
Consider the Taylor expansion of $f$
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