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If two power series $$f_n(x)=\sum_{n=0}^{\infty}a_nx^n,\space g_n(x)= \sum_{n=0}^{\infty}b_nx^n$$ converge on the same interval $(-R,R), \space (R>0)$ to the same function $f$, then we have to prove $a_n=b_n$

Therefore both of $f_n(x) \space and \space g_n(x)$ has positive and same radius of convergence $R$ and both of them are converging in the same function $f$.

Hence, $f'_n(x)=g'_n(x)\\ \therefore f'(x)=g'(x)\\ \Rightarrow f'(0)=g'(0)\Rightarrow a_1=b_1$, and $f''(0)=g''(0)\Rightarrow a_2=b_2$ and obviously $f(0)=g(0)\Rightarrow a_0=b_0$. So from here can I conclude that $a_n=b_n$?

Any help is highly appreciated.

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First, better rename both functions $f,g$, omitting the $n$ since it is just a dummy index. Compare the nth coefficients of the differentials $f^{(n)}$ and $g^{(n)}$ evaluated at $0$ and you get $a_n=b_n$ using $\frac{d}{dx}\sum_{\infty}..= \sum_{\infty}\frac{d}{dx}..$ with holds on their convergence domain.

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