Uniqueness Of A Power Series

Let there be two power series which are equal for all \(x\) in an interval containing \(a\):

\[\sum^{\infty}_{n=0} c_n (x-a)^n = \sum^{\infty}_{n=0} d_n (x-a)^n\]

Let the series be equal to \(f(x)\):

\[\begin{align} f(x) &= c_0 + c_1 (x-a) + c_2(x-a)^2 + c_3(x-a)^3 + c_4(x-a)^4 + \cdots \\ &= d_0 + d_1 (x-a) + d_2(x-a)^2 + d_3(x-a)^3 + d_4(x-a)^4 + \cdots\end{align}\]

This means, when \(x=a\), \(f(a) = c_0 = d_0\). If we differetiate the power series, we get:

\[\begin{align} f'(x) &= c_1 + (2)c_2(x-a) + (3)c_3(x-a)^2 + (4)c_4(x-a)^3 + \cdots \\ &= d_1 + (2)d_2(x-a) + (3)d_3(x-a)^2 + (4)d_4(x-a)^3 + \cdots\end{align}\]

This means, when \(x=a\), \(f'(a) = c_1 = d_1\). If we differetiate again, we get:

\[\begin{align} f'(x) &= (2)c_2 + (3)(2)c_3(x-a) + (4)(3)c_4(x-a)^2 + \cdots \\ &= (2)d_2 + (3)(2)d_3(x-a) + (4)(3)d_4(x-a)^2 + \cdots\end{align}\]

This means, when \(x=a\), \(f''(a) = c_2 = d_2\). Using a similar argument, we can claim \(c_i = d_i\) for all integer \(i \ge 3\).

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