INITIAL INQUIRY
Does this inequality have any solutions over the positive integers?
$$\dfrac{q^{k + 1} - 1}{q^k (q - 1)} < \dfrac{2q}{2(q - 1) + (1/a)^2}$$
I tried asking WolframAlpha for (some of) the solutions, it seems that there is no solution where all of $q, k$, and $a$ are positive integers.
MY ATTEMPT
Suppose that there exist $q, k, a \in \mathbb{Z}^{+}$ such that $$\dfrac{q^{k + 1} - 1}{q^k (q - 1)} < \dfrac{2q}{2(q - 1) + (1/a)^2}.$$
First, I noted that $$\dfrac{q^{k + 1} - 1}{q^k (q - 1)} < \frac{q}{q - 1}$$ and that $$\frac{q}{q - 1}$$ is the least upper bound of the set $$\mathscr{Q} = \{\dfrac{q^{k + 1} - 1}{q^k (q - 1)}\}.$$
In particular, I also noticed that $$\dfrac{2q}{2(q - 1) + (1/a)^2} < \frac{q}{q - 1}.$$
This contradicts the fact that $q/(q - 1)$ is the least upper bound of $\mathscr{Q}$.
Hence, the inequality $$\dfrac{q^{k + 1} - 1}{q^k (q - 1)} < \dfrac{2q}{2(q - 1) + (1/a)^2}$$ has no solutions where all of $q, k$, and $a$ are positive integers.
FINAL INQUIRY
Is my solution correct? If it is incorrect, how can the proof be mended so as to produce a valid argument?
Actually there are infinitely many positive interger solutions.
Consider $q=2,k=1$ and $a\in\mathbb Z\cap[2,\infty)$, then $$\frac{q^{k+1}-1}{q^k(q-1)}=\frac32<\frac4{2+(1/2)^2}\leq\frac{2q}{2(q-1)+(1/a)^2}.$$
The issue in the proof: The left-hand side is $\frac{q-1/q^k}{q-1}$, so you are right that $\frac q{q-1}$ is the least upper bound for all $q>1,k>0$. But no matter how large $q$ is, there always exists some gap between $\frac{q-1/q^k}{q-1}$ and $\frac q{q-1}$, i.e. the interval $(\frac{q-1/q^k}{q-1}, \frac q{q-1})$ is non-empty, and the smaller the $q$ the larger the gap. On the other hand, the right-hand side $\frac{2q}{2(q-1)+(1/a)^2}$ can get arbitrarily close to $\frac q{q-1}$ if $a$ is sufficiently large, this means we can choose a small $q$ such that the gap $(\frac{q-1/q^k}{q-1}, \frac q{q-1})$ is big, and choose a large $a$ such that the right-hand side lies in this gap, i.e. $\frac{2q}{2(q-1)+(1/a)^2}\in(\frac{q-1/q^k}{q-1}, \frac q{q-1})$. By this idea, we can construct infinitely many positive integer solutions.