Is it true that if $a_0 1$, then the series $\sum_{n=0}^\infty \frac{1}{a_n \cdot a_{n+1}}$ is irrational?

Erdős problem 1051

Is it true that if $a_0 < a_1 < a_2 < \cdots$ is a strictly increasing sequence of integers with $\liminf a_n^{1/2^n} > 1$ , then the series $\sum_{n=0}^\infty \frac{1}{a_n \cdot a_{n+1}}$ is irrational?

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Problem row: sha256:14d7770476e44ab5c9b1fa591a131f4cdea4230399a4f93038f991e714a6269b
Metadata: sha256:54b8d7bdf7f0f27ea9ee7cae38a70241d975b79940058b1f91fd4eddf8c09164
Observation: sha256:9c8326e865b6017b103af7611c852a06776d81964979b8ab5a1b800cd550dff6
Content: sha256:3c12ad29c9d4620d22c8face4db77c75923f438c59c81bbca5c8d4aba575219b

Repository: sha256:a956b84c437202e5a02cc9e036a621bd14a302b34a75758115730bdbb77c52a4
Projection: sha256:a4a96441668a9f8b33afbcb3f696a2639f5fae145f36ddc063d7f67d5fb87ce8
Source commit: 36c6d0fef71a3fec84d9dcd5eeca4e22e378f7cb