In the book "p-adic Numbers, An Introduction" by Gouvêa he gives in his informal introduction the calculation (modulo typo)
$\frac{X}{X-1}=\frac{2+(X-2)}{1+(X-2)}=\color{red}{2}-(X-2)+(X-2)^2-(X-2)^3\pm\dotso$.
Problem 1 asks to verify this, but this should be simply wrong. I do not see where the $2$ should come from.
We get:
$\dfrac{2+(X-2)}{1+(X-2)}=\color{red}{\dfrac{2}{1+(X-2)}}+(X-2)\dfrac{1}{1+(X-2)}=\dfrac{2}{1+(X-2)}-(X-2)+(X-2)^2\pm\dotso$
as $\frac{1}{1+(X-2)}=\sum_{n=0}^\infty (-(X-2))^n$
Or am I just mistaken?
Thanks in advance.
No, the book is correct.
It's analogous to polynomial long division, but in reverse order.
In other words, at each stage, divide the leading monomial of what remains of the dividend by leading monomial of the divisor, then muliply and subtract. Repeat forever.