Question regarding Power Series

1. May 21, 2015

Potatochip911

1. The problem statement, all variables and given/known data
It is stated in my textbook that the sum $\sum_{0}^{\infty} 8^{-n}(x^2-1)^n$ is not a power series but can be turned into one using he substitution $y=x^2-1$ which then becomes the power series $\sum_{0}^{\infty} 8^{-n}y^n$ They aren't offering any explanation as to why and I have evaluated the interval on which the series converges and it gives the same result regardless of whether or not the substitution is used. I guess what I'm wondering is why isn't the first sum a power series?

2. Relevant equations

3. The attempt at a solution

2. May 21, 2015

Jazzman

The x2 inside the parenthesis makes it not a power series. A power series must have x1 inside the parenthesis.

3. May 21, 2015

Potatochip911

Okay thanks for the information.

4. May 22, 2015

Orodruin

Staff Emeritus
Well, technically it is a power series in $x^2-1$.

5. May 22, 2015

Ray Vickson

No. A series of the form $\sum c_n x^{2n}$ IS a power series. The reason the series $\sum_n 8^{-n}(x^2-1)^n$ is not a power series (in $x$) is that it takes powers of a multi-term polynomial of $x$, rather than of a monomial in $x$. That is, the function $x^2-1$ is not a monomial.