This theorem says that if for two power series, the input always gives the same output, then the coefficients of the power series are the same for both.
How can we prove this?
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consider the difference between the two series - which will be the constant function $d(x)=0$ - expressed as a power series all the coefficients of $d(x)$ are $0$ – WW1 Jun 09 '17 at 19:59
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Of course to have an "output" for any value of $x$ other than $0$, the series must have positive radius of convergence, so that must be assumed. Then Taylor's theorem relates the coefficients to the values of the "output" and its derivatives at $x=0$.
Robert Israel
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