Could I prove that the ratio test still works using $\limsup(\frac{a\_{n+1}}{a\_n})$ instead of $\lim(\frac{a\_{n+1}}{a\_n})$?
I think for $\limsup<1$ I could show that for $\epsilon>0, N>1 \limsup(\frac{a\_{n+1}}{a\_n})<1-\epsilon.$ From there I can solve that $\lvert a\_k\rvert<(1-\epsilon)^{k-N}\lvert a\_N\rvert$ for $k>N$. Thus, by comparison test the left-hand side will converge absolutely. Is this enough to show I can use the ratio test with $\limsup$?