Finding linear convolution of two time series
There is a small mistake in your solution. Here $f_k$ and $g_k$ are infinite duration signal, ie. $n \to \infty$. You made a mistake that you took  n in $h_n$ and number of coefficients in $f_k$ or $g_k$ as same, but it is different.

$$h_n = \sum_{k=0}^{\infty} f_k g_{n-k} = \sum_{k=0}^{\infty} a^k b^{n-k} = b^n \sum_{k=0}^{\infty} \left(\frac{a}{b} \right)^k$$

$$h_n = b^n \cdot \frac{1}{1 - \frac{a}{b}}$$

$$h_n = \frac{b^{n+1}}{b-a}$$

If you want to finite the number of coefficients in $f_k$ and $g_k$, use other variables other than n.

Then n in the RHS of your equation will replace with the new variable.