I've been reading through other questions on this site and external resources for a few hours now but seem to be having a mental block, probably through some elementary misunderstanding of my course notes or most likely because my mathematical maturity is certainly prepubescent! I haven't a clue how I would show the following (I know it's ludicrously simple):
If there is a sequence $\{a_{k}\}=(-3)^{k}$, what is its generating function?
I'm not quite sure what is 100% going on behind a generating function. Any "for dummies" explanation would be greatly appreciated.
Thanks for your patience and interest.
For the gf:
$$G (x)=\frac{1}{1+3 x}$$
$$ \frac{1}{1-a x}=\sum _{k=0}^{\infty } a^k x^k $$
You can get that from any table of generating functions.
https://en.wikipedia.org/wiki/Generating_function
Or you can derive that one from the simplest one,
$$ \frac{1}{1- x}=\sum _{k=0}^{\infty } x^k $$
by substituting x = ax in it. So you see there are at least 3 ways, 1) sum the series, 2) Look it up in a table and 3) derive it from one we already know.
Try Applied Combinatorics by Tucker starting with chapter 6. Best way to learn anything at all is to see what it can be used for.
For your comments below:
$$s = 1 + ax + a^2x^2 + a^3x^3 +...+$$
$$axs = ax + a^2x^2 + a^3x^3 +...+$$
$$s - axs = 1$$
$$s(1-ax)=1$$
$$ s= \frac{1}{1-ax}$$