Question regarding Power Series

[Potatochip911]

Homework Statement

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?

Homework Equations

The Attempt at a Solution

 

[Jazzman]

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

 

[Potatochip911]

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

Okay thanks for the information.

 

[Orodruin]

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

 

[Ray Vickson]

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

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.