# Could the series $\sum (x-3)^n/n$ be seen as a power series if we consider $1/n$ as $c_n$?

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 $$c_n(x-a)^n$$?

• Taking $a$ to be $3$. The definition is here en.wikipedia.org/wiki/Power_series – Winther Apr 16 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 at 16:46
• math.stackexchange.com/questions/392918/… 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 at 17:01

## 1 Answer

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