Show that 1 x x2 is a unit in Qx and express 11 x x2 as

Show that 1 - x - x^2 is a unit in Q[[x]], and express 1/(1 - x - x^2) as a power series.

Solution

Any element in Q[[x]] (the ring of formal power series ) is invertible , provided it has a non-zero constant term.

Hence 1-x-x² is invertible in Q[[x]].

[1-x-x² ]^-1 = [1-(x+x² )]^-1 = 1+(x+x²) + (x+x²)² +......(x+x²)ⁿ +.........(By considering either geometric series or binomial theorem)

is the power series expansion (and hence the inverse of 1-x-x² in Q[[x]], in fact the coefficients are in Z)