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Being new to power series just a little confusion here on the Power series use of it's coefficient. As n goes from 0 to infinity or 1 to infinity as in indicated in the variable x the same n is also indicated in it's coefficient.

I noticed that the examples use n not only to indicate it's position in the chain but also the n is used to determine the value of the coefficient such as ( -1) ^n being the coefficient ....my question is ...is this always the case that the n acts also as a function? Must is always be dual purpose or can it be just a simple place holder ? Thank you

Sedumjoy
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    Um ... if the coefficients could not be a function of $n$, all you could do would be pure geometric series, which would make the concept of power series rather uninteresting. – hmakholm left over Monica Jan 13 '18 at 02:00
  • Ok...that helps me...but doesn't the power series become a geometric series once a value for x is chosen ? The other puzzle to me is that I assume the power series is not a ratio from term to term but is this true? – Sedumjoy Jan 13 '18 at 02:09
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    No, the coefficients (generally, as @HenningMakholm points out) stop it from being a simple geometric series. – Chris Jan 13 '18 at 02:16

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The coefficients do not have to be a function of $n$. For example, you could have $a_n=2$ for every $n$, then your series is: $\sum_{n=1}^{\infty}{2x^n}=2\sum_{n=1}^{\infty}{x^n}$.

Sort of Damocles
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