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I'm just trying to be sure I understand power series correctly. Would the series $\sum \frac{(x-3)^n}{n}$ be seen as a power series if we consider $\frac 1n$ as $c_n$, seeing as (taking $a$ here to be zero) the formula for a term of a power series is $c_n(x-a)^n$?

James Ronald
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    Taking $a$ to be $3$. The definition is here https://en.wikipedia.org/wiki/Power_series – Winther Apr 16 '19 at 16:11
  • @Winther Thank you for the help! Yes I see now that $a=3$ would make more sense. Also I didn't think to check Wikipedia but I encountered Borel's theorem which is interesting! So is every geometric series a Maclaurin series? – James Ronald Apr 16 '19 at 16:46
  • https://math.stackexchange.com/questions/392918/are-taylor-series-and-power-series-the-same-thing A geometrical series is a series on the form $\sum_{n=n_1}^{n_2} ax^n$ which you can show equals $a(x^{n_2+1}-x^{n_1})/(1-x)$ and is also the Maclaurin series of this function. – Winther Apr 16 '19 at 17:01

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Yes you can consider it that way, and in fract it is related to the Taylor series for $$ \log\left( \frac{1}{4-x}\right) $$

Mark Fischler
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