Let $\{a_k\}$ be a sequence of non-zero real numbers and suppose that
$p = \lim_{k \to \infty} k\left(1-\frac{|a_{k+1}|}{|a_k|}\right)\quad\text{exists}$
Prove that $\sum_{k=1}^{\infty}a_k$ converges absolutely when $p > 1$.
I've tried manipulating the equation to isolate $\frac{|a_{k+1}|}{|a_k|}$ and use the Ratio Test. But it doesn't seem to work because then $\lim_{k \to \infty} \frac{|a_{k+1}|}{|a_k|} = 1$, and the Ratio Test is inconclusive.
Probably some other Test (like the Logarithmic Test?) needs to be used but I'm unsure how.
Any advice would be appreciated. Thanks.