The question asks
Represent the function $f(z) = \frac{z+1}{z-1}$ by its Maclaurin series for $|z|<1$ and its Laurent series for $1<|z|<\infty$.
The answers the book gives are:
Maclaurin series: $-1-2\sum_{n=1}^{\infty}z^n$
Laurent series: $1+2\sum_{n=1}^{\infty}\frac{1}{z^n}$
The answers I got were:
Maclaurin series: $-(z+1)\sum_{n=0}^{\infty}z^n$
Laurent series: $\frac{z+1}{z}\sum_{n=0}^{\infty}\frac{1}{z^n}$, for the same intervals.
Where do the $1$ and $2$ constant/coefficient come from? Are my answers correct but just not in the given form? What am I missing?
Hint. Your answer for the Maclaurin series can be expanded: $$\eqalign{-(z+1)\sum_{n=0}^\infty z^n &=-(z+1)(1+z+z^2+\cdots)\cr &=-z-1-z^2-z-z^3-z^2-\cdots\ .\cr}$$ As you can see, there is often more than one $z$ term with the same exponent. If you collect terms you will get the simplified series.