you need to be clearer about what Q[[t]] is. usually, this notation means all formal power series, not polynomials (a polynomial is a "finite power series").

it is true that all units in Q[t] (the ring of rational polynomials) are elements of Q*.

but let's consider what we have in power series.

consider the power series:

1 + t + t^{2}+ t^{3}+...+ t^{n}+.....

what happens when we multiply this by 1 - t?

(1 - t)(1 + t + t^{2}+ t^{3}+...+ t^{n}+.....)

= (1 + t + t^{2}+ t^{3}+...+ t^{n}+.....) - (t + t^{2}+ t^{3}+...+ t^{n}+.....) = 1 + 0t + 0t^{2}+ 0t^{3}+.... = 1.

more succintly:

this is not a statement of convergence, we are not considering "values" for t, it is a statement about FORMAL power series.

the general problem of determining the units in Q[[t]] is a bit more complicated. if

is a unit of Q[[t]], there is some element:

with ur = 1.

we have, for the product:

, where

if this is to be equal to 1, then c_{0}= 1, c_{k}= 0, for k > 0.

now c_{0}= a_{0}b_{0}, so we see a_{0}cannot be 0, if u is to be a unit. if a_{0}≠ 0, define the b_{k}like so:

.

verify that this definition makes r an inverse for u.