uniqueness of power series
Could one and the same function be written as two genuinely different power series around the same center? Reassuringly, no. A function pins down its power-series coefficients completely — there is exactly one way to expand it. This is what makes coefficient-matching a legitimate technique: if two power series represent the same function, you may equate them term by term.
Statement: if sum a_n (x - c)^n = sum b_n (x - c)^n for all x in some interval around c (or even on a set with c as a cluster point), then a_n = b_n for every n. The proof is immediate from the coefficient formula: both sides have the same sum function f, and each coefficient must equal f^{(n)}(c)/n!, so they coincide. Equivalently, the only power series summing to the zero function is the one with all coefficients zero.
A sharper version underlies the identity theorem for analytic functions: if a power series with positive radius vanishes on a set having a cluster point inside the disk, all its coefficients are zero. This rigidity is dramatic — agreeing on a tiny cluster of points forces two analytic functions to agree everywhere they are both defined and connected, a property false for merely smooth functions.
If a + bx + cx^2 + ... = 0 for all x in (-r, r), then setting x = 0 gives a = 0; differentiating and setting x = 0 gives b = 0; and so on, forcing every coefficient to vanish. There is no nonzero way to sum to the zero function.
Only the zero series sums to the zero function.