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How to show the coefficients of power series are the same for two power series that coincide on an interval. I first thought about using $x=0$ to prove $a_0=b_0$ and using $n$th derivatives to show $a_n=b_n.$ But I realized that $0$ is not necessary in the interval of coincidence. Any hints will be appreciated!

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    $\begingroup$ can't you do the same, but consider a power series expanded about $x_0$, and consider the function and all derivatives at $x=x_0$. $\endgroup$ – Ellya Apr 3 '15 at 13:35
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    $\begingroup$ I guess the question is like this. Suppose two power series centered at zero have radii of convergence ${}\ge 4$, and the values of the two series coincide on the interval $(3,4)$. Show that the coefficients of the two power series agree. A deleted answer used complex analysis for this. Is there a more elementary way? $\endgroup$ – GEdgar Apr 3 '15 at 14:40
  • $\begingroup$ That is the question. $\endgroup$ – user228492 Apr 3 '15 at 19:44
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Power series are defined for other values than $0$, so as long as you do the expansion around $x_0$ in the interior of your interval (not on the boundary), it will work.

The theorems are then the same, and you can use the relation between $n$-th derivative and $n$-th coefficient to conclude, as you suggested.

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