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google：
series Σ _{k＝0,1,…,+∞}(1/(k!＋1)) irrational number proof

AIモード 回答
The question of whether the series Σ _{k＝0,1,…,+∞}(1/(k!＋1)) is irrational appears to be an open problem in mathematics. No established proof has been found to confirm its irrationality, though it is strongly suspected to be an irrational number. 
以下略
(引用終り)

で、”irrational appears to be an open problem in mathematics. No established proof has been found to confirm its irrationality”
とでる

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