Let $a_k≠0, \forall k \in \mathbb{N}$. If $$\limsup_{k\to\infty}\left|\frac{a_{k+2}}{a_k}\right|<1$$ then the series $\sum_{k=1}^\infty a_k$ is absolutely convergent.
I am really interested into seeing a proof of this but I cannot find one. For the original ratio test, there are many proofs but this one is different. Has anyone an idea? Thanks.
On
The condition $$\limsup_{k\to\infty}\left\vert\frac{a_{k+2}}{a_k}\right\vert<1$$ implies that it exists $N \ge 1$ and $0 \le K \lt 1$ such that
$$\left\vert\frac{a_{k+2}}{a_k}\right\vert \le K$$ for all $k \ge N$. Therefore
$$\left\vert\frac{a_{N+2p}}{a_N}\right\vert \le K^p, \left\vert\frac{a_{N+2p+1}}{a_{N+1}}\right\vert \le K^p$$ for all $p \ge 0$ and
$$\begin{aligned}\sum_{k \ge N} \vert a_k \vert &= \sum_{k \ge N\\k\textrm{ odd}} \vert a_k \vert + \sum_{k \ge N\\k\textrm{ even}} \vert a_k \vert\\ &\le \vert a_{N+1} \vert\sum_{k \ge 0} K^{p} + \le \vert a_{N} \vert\sum_{k \ge 0} K^{p}\\ &\le \frac{2}{1-K} \max(\vert a_N \vert, \vert a_{N+1} \vert) \end{aligned}$$
Proving that $\sum a_k$ converges absolutely.
To follow leoli1's comment, you can do it as follows.
By the assumption and directly applying the ratio test you can show that the following two series are convergent: $$ \sum_{k=1}^\infty |a_{2k}|,\quad \sum_{k=1}^\infty |a_{2k+1}| $$
Now the partial sum of the original (absolute) series has the following estimate by the triangle inequality $$ \sum_{k=1}^n|a_k| \le \sum_{k=1\\k\textrm{ odd}}^n|a_k| +\sum_{k=1\\k\textrm{ even}}^n|a_k|\tag{1} $$ Taking $n\to\infty$, we are done.
Notes. One actually has equality in (1) and thus the triangle inequality is not needed. (Thanks to hardmath's comment below.)