Question

Is the geometric series a power series?

Answer 1

No. A geometric series is one where each succesive term is produced by multiplying the previous term by a constant, r; something like\[\sum_{n=1}^{\infty}a _{1}r^{n-1}\]A power series is a series involving powers of the variable (not a constant); say something like\[\sum_{x=0}^{\infty}a _{n}(x-c)^n\]

Answer 2

My book says that a geometric series can be written as a power series \[\sum_{n=0}^{\infty}{x^n}\]

Answer 3

Recall that the geometric series is convergent when -1<r<1:\[1+r+r^2+r^3+. . .=\frac{1}{1-r}\]If we rename r to x and transpose sides\[\frac{1}{1-x}=1+x+x^2+x^3+x^4+. . .\]when -1<x<1. We can rewrite as a power series. that is what your book must be referring to. In that sense, virtually any series can be converted into a power series.