The z transform and first principles

[nothing909]

Homework Statement

shown in the picture is the question and answer to it, but i don't understand how they're getting it.

this is me just not understanding the maths and i know its not difficult but I've been stuck on it for a while, so can someone explain in detail how you get the last two lines of the equation.

 

[anorlunda]

Our rules say that you must show your attempt at solution before helpers are allowed to help. Just saying, "I don't get it." is not enough. Your textbooks and lectures must have given you some foundation needed for this problem.

 

[nothing909]

They do show me how to get the first part, but not how to converge. I think the last line you multiple by denominator and numerator by z squared, but that’s all I got.

All I need is a quick explanation of how they are getting the last 2 lines of the equation, I understand the rest of it

 

[StoneTemplePython]

This text looks... sloppy. Two glaring problems:

1) The inequalities at the top of the page don't match the summation -- i.e. the top specifically states that $a_0 = 0$ and $a_{N-1} = 0$, so why are they included when moving from a formal infinite series to a finite one? I'm fairly confident that they meant to write $0 \leq n \leq N-1$ and didn't pay attention to strictness of inequalities.

2) Convergence of the geometric series really has nothing to do with the result at the bottom of the page. It's just telescoping a finite geometric sum -- a technique that is frequently taught in middle school or high school. The result holds whether or not you're in the radius of convergence for a geometric series.

That's a lot of issues for a very short extract. If there's an errata sheet, get it and include relevant sections with these posts. If there isn't an errata sheet, I'd get a different text or recognize that learning from this will leave you, at best, as confused as the authors.