How to show that partial sums of an infinite series can be written in a different form

Question

I'm working through a book over Christmas break and there is an exercise in which it gives the geometric series $S = 1+z+z^2+\dots+z^n+\cdots$, and asks to show that the sum of the first $n+1$ terms can be written

$$S_{n+1}=\frac{z^{\frac{n+1}{2}}-z^{-\frac{n+1}{2}}}{z^{\frac 1 2} - z^{-\frac{1}{2}}}n^{\frac n 2}$$

Can someone provide me with some insight as to how to go about solving this?

Answer 1

There is a typo, it should be $$\frac{z^{(n+1)/2}-z^{-(n+1)/2}}{z^{1/2}-z^{-1/2}}z^{n/2}.\tag{1}$$ For writing the $z^{n/2}$ at the end as $\frac{z^{(n+1)/2}}{z^{1/2}}$, and absorbing these powers of $z$ into the top and bottom of (1), we get $$\frac{z^{n+1}-1}{z-1},$$ which, for $z\ne 1$, is a familiar expression for the sum of a finite geometric series. For the modified expression, we also need to exclude the uninteresting case $z=0$.