Showing that a generating function is equivalent to some fraction

Question

I am working with generating functions and am required to prove that the generating function for the sequence $\{a_k\}$ where $a_k = (-8)^k$ for all integers $k\geq0$ is $\cfrac{1}{1+8x}$ and I have gotten it to:

$$g(x)[1+8x] = \lim \limits_{n\to\infty} 1 - (-8)^{n+1}x^{n+1}$$

I can see that I want to take $(-8)^{n+1}x^{n+1}\to0$, but I don't know how I can say this is alright.

It works for $0\lt x\lt\lt\lt1$ I suppose, but I don't see how I can justify this.

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This has been shown to me. It is a very similar question, and they have gotten to teh stage where $g(x)-xg(x)=1$, this is what I want, but this ugly one above can only reduce if $x\lt\frac18$. The current only answer says that I can't assume this, and I feel as though that is true. But his formula assumes what he is saying I can't.

Answer 1

When using generating functions we treat the series $$\sum_{n=0}^{\infty}a_nx^n$$ as formal expression, and we dont care about convergence rate and stuff like that.

The notation $\frac{1}{1+8x}$ is nothing but a simple notation, and we dont actually care about subtituting x with anything, just with the cofficient of $x^k$ in the series.

For your series the generating function is $$f(x) = \sum_{n=0}^{\infty}(-8)^nx^n = \sum_{n=0}^{\infty}(-8x)^n = \frac{1}{1+8x}$$

Where the last part is that notation that is convinient since its the sum of the geometric series

Answer 2

Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations defined for formal power series. These expressions in terms of the indeterminate x may involve arithmetic operations, differentiation with respect to x and composition with (i.e., substitution into) other generating functions; since these operations are also defined for functions, the result looks like a function of x. Indeed, the closed form expression can often be interpreted as a function that can be evaluated at (sufficiently small) concrete values of x, and which has the formal power series as its Taylor series; this explains the designation "generating functions".

However such interpretation is not required to be possible, because formal power series are not required to give a convergent series when a nonzero numeric value is substituted for x. Also, not all expressions that are meaningful as functions of x are meaningful as expressions designating formal power series; negative and fractional powers of x are examples of this.

http://en.wikipedia.org/wiki/Generating_function