Uniqueness of power series
Theorem 7.1 (Uniqueness of power series). If \(\sum a_n (z - z_0)^n\) converges to \(f(z)\) on some disc \(\abs {z - z_0} < r\), then
\[ a_n = \frac {f^{(n)}(z_0)}{n!} \qquad (n \geq 0). \]
In particular, two power series with the same sum near \(z_0\) have the same coefficients.
Source: [1, ch. 5]
Proof. Applying Theorem 6.1 \(k\) times,
\[ f^{(k)}(z) = \sum _{n=k}^{\infty } \frac {n!}{(n-k)!} \, a_n (z - z_0)^{n-k}, \qquad \abs {z - z_0} < r. \]
Putting \(z = z_0\) kills every term except \(n = k\), so \(f^{(k)}(z_0) = k! \, a_k\). If two series have the same sum \(f\) near \(z_0\), both sets of coefficients equal \(f^{(n)}(z_0)/n!\).