Puzzling Trinomials and validity

Question

A textbook exercise asked me to state the set of values of $x$ for which $1-x+x^3-x^4 +x^6-x^7...=(1+x+x^2)^{-1}$ is a valid expansion. The LHS is the sum of two infinite geometric progressions, both of which are valid for $|x|<1$ and this is the answer given in the back of the book. However, isn't the RHS is valid for $|x + x^2|<1$, by the binomial theorem? Sketching the graph of this function would give a different set of values of $x$. Can anyone clear up this confusion?

Answer 1

Write: $1(1-x)+x^3(1-x)+x^6(1-x)+...=(1-x)(1+x^3+x^6+...)=\dfrac{1-x}{1-x^3}=\dfrac{1}{1+x+x^2}$. This equation is true if $|x|<1$.