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.