Assume $k < 1$ . By definition $\exists N$ such that
$n > N \implies |\frac{a_{n+1}}{a_n} - k| < \frac{1-k}{2} \implies |\frac{a_{n+1}}{a_n}| < \frac{1+k}{2} < 1$

i.e. eventually the series $|a_n|$ becomes less than a convergent geometric series, therefore a shifted subsequence of $|a_n|$ converges by the comparison test. Note that a general sequence $b_n$ converges iff a shifted subsequence of $b_n$ converges. Therefore, by the absolute convergence theorem, the series $a_n$ converges.

Similarly for $k > 1$ a shifted subsequence of $|a_n|$ becomes greater than a geometric series tending to $\infty$ , and so also tends to $\infty$ . Therefore $a_n$ diverges.