In a power series is it necessary that the coefficient be a function of n as n increases or just a place holder? 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

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: $\sum_{n=1}^{\infty}{2x^n}=2\sum_{n=1}^{\infty}{x^n}$.