PROBLEM
Two real, causal time series
$$f_k = a^k \quad \text{and} \quad g_k = b^k$$
where $a \neq b$, $|a| < 1$, and $|b| < 1$, are given for $k = 0,1,2,...$
Find the linear convolution $h_n$ of the time series $f_k$ and $g_k$.
ATTEMPTED SOLUTION
I think perhaps my attempted solution here can be improved upon. We have from the definition:
$$h_n = \sum_{k=0}^{n} f_k g_{n-k} = \sum_{k=0}^{n} a^k b^{n-k} = b^n \sum_{k=0}^{n} \left(\frac{a}{b} \right)^k$$
If we let $R = \left(\frac{a}{b} \right)$, the last term can be written as:
$$h_n = b^n \cdot \frac{1 - R^{n+1}}{1 - R}$$
or
$$h_n = \frac{b^n - \frac{a^{n+1}}{b}}{1 - \frac{a}{b}}$$
$$h_n = \frac{b^{n+1} - a^{n+1}}{b-a}$$
I don't see how I can improve upon this answer though, but since we are given certain conditions in the problem, I have a feeling that it is possible to get a more elegant solution. I see that as $n \to \infty$, $h_n \to 0$, but is there any way I can "improve" on the answer for any $n$?