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Unfortunately, I don't have much detail to give here. But is the general idea to cancel out the constant obtained from taking the derivative.

For instance, say my function was $f(x)=f_o+f_1x+f_2x^2+...$

Then $f'(x)=f_1+2f_2x+...$

And if the expansion is centered around $x=0$...
$f'(0)=0$,
$f''(0)=2f_2$
$f'''(0)=3*2f_3$

Therefore
$f_0=f(0)$
$f_1=f'(0)/1$
$f_2=f''(0)/2$

And so forth. Is that where the factorial comes from?

It is quite clear for a polynomial, but what about a trig function such as $sin(x)$ other than using taylor's formula?

  • 0
    Sure. Differentiate $x^n$ a total of $n$ times, or integrate $1$ a total of $n$ times.2012-10-08
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    If that's the case, how can a trig function be explained?2012-10-08
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    Sine and cosine have (reciprocal) factorials. Everybody does. The examples of missing, or apparently missing factorials are things like $1/(1-x)$ and its relatives like $\log(1+x)$ and $\arctan x$, where factorials are prouced by the differentiation process and largely cancel the factorial that comes from $x^n$.2012-10-08
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    Ignoring differentiability issues and rigor, you can obtain the coefficients in a purely algebraic manner by following the method I used in my answer at [power series expansion](http://math.stackexchange.com/questions/178704/power-series-expansion).2012-10-08

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