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!
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.