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Let $\delta, \epsilon \in \mathbb{R}$, $\delta >0$, $\epsilon >0$. Let $\{ a_k\}^\infty$,$\{ b_k\}^\infty$ be sequences of positive integers such that

$\lim \sup_{k \rightarrow \infty} \frac{a_{k+1}}{ (\prod^k a_j)^{2 +2/\epsilon + \delta}} \frac{1}{b_{k+1}} = \infty$ and for every suf. large $k$ we have $\sqrt[1+\epsilon]{ \frac{a_{k+1}}{b_{k+1}} }\geq \sqrt[1+\epsilon]{ \frac{a_{k}}{b_{k}} }+1$. Why is the infinite series $\sum^\infty \frac{b_k}{a_k}$ summable? (This claim comes from a theorem in a published paper whose proof we are formalising using the proof assistant Isabelle/HOL. Summability easily follows if we strengthen the assumption substituting $\lim \sup$ with $\lim$, but is there a way to show summability without strengthening any assumption?) Thank you in advance!

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The condition that for all large enough $k$ we have $\sqrt[1+\epsilon]{ \frac{a_{k+1}}{b_{k+1}} }\geq \sqrt[1+\epsilon]{ \frac{a_{k}}{b_{k}} }+1$ implies that $\sqrt[1+\epsilon]{ \frac{a_{k}}{b_{k}} }>k-c$ and hence $\frac{b_{k}}{a_{k}}<\frac1{(k-c)^{1+\epsilon}}$ for some real $c>0$ and all large enough $k>c$, which yields the convergence of series $\sum^\infty \frac{b_k}{a_k}$.

(The condition $\lim \sup_{k \rightarrow \infty} \frac{a_{k+1}}{ (\prod^k a_j)^{2 +2/\epsilon + \delta}} \frac{1}{b_{k+1}} = \infty$ is not needed here.)

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  • $\begingroup$ Dear Iosif, many thanks - of course your hint will be acknowledged in our code of the formal proof! $\endgroup$ – Angeliki Koutsoukou Argyraki Feb 14 at 23:50

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