Infinite series question with z-transform addendum

[kostoglotov]

Homework Statement

Hello,

I am currently doing some holiday pre-study for signals analysis coming up next semester. I'm mainly using MIT OCW 6.003 from 2011 with some other web resources (youtube, etc).

The initial stuff is heavy on the old infinite series stuff, that seems often skimmed over in previous calculus study, and was for me.

Working from a transfer function as an infinite series, I was curious if I could figure out a closed-form of the infinite right hand sum (0 to inf) of $nR^{n-1}$

$$0+1+2R+3R^2+4R^3+5R^4+...$$

by working in reverse through parts of synthetic division I found that $\frac{1}{(1-R)^2}$ works for a closed form of the previous infinite series.

Then by the same technique found that an infinite right hand sum for $n^2a^n$ has the closed form $\frac{a(1+a)}{(1-a)^3}$.

I thought I saw a pattern ($\frac{a(1+a)^k}{(1-a)^{k+1}}, from \ n^ka^n$), but for the infinite right hand sum of $n^3a^n$ reversing through synthetic division reveals a closed form $\frac{a(1+4a+a^2)}{(1-a)^4}$...so I thought maybe the coefficients of the numerators polynomial are simply getting squared...but going through this again for $n^4a^n$ shows that this is not the case.

What is the general closed form formula for an infinite right hand series of $n^ka^n$?

Also, I am aware of the property of Z-Transform Differentiation, $nx[n] \leftrightarrow -z\frac{dX(z)}{dz}$. Does this relate to what I'm exploring above? If so, how?

PS. I'd love to know of any other good resources for self-teaching signals and systems stuff :)

Homework Equations

The Attempt at a Solution

 

[MisterX]

I don't think there is a general closed form with elemental functions. You seem to be asking about the poly-logarithm function just shifted by 1.