I'm thinking that since $\sum a_n x^n$ = $\sum b_n x^n$ are equal, then it should be obvious that $a_n = b_n$ for all $n\in\mathbb{N}$. Is this true or false? Thank you!
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Please don't use $*$ for multiplication. – zhw. Oct 21 '19 at 21:03
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Have a look here: https://math.stackexchange.com/questions/856932/can-there-be-more-than-one-power-series-expansion-for-a-function – Théophile Oct 21 '19 at 21:22
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It's true when ${x: \sum_n a_n x^n = \sum_n b_n x^n}$ has a limit point. – Oct 21 '19 at 21:27