Is 1 11 </mfrac> + 1 111 </mfrac> + 1 1111

juanberrio8a

juanberrio8a

Answered question

2022-06-09

Is 1 11 + 1 111 + 1 1111 + an irrational number?

Answer & Explanation

Rebekah Zimmerman

Rebekah Zimmerman

Beginner2022-06-10Added 32 answers

If we define the "Lambert series"
f ( x ) = n = 1 x n 1 x n = n = 1 1 ( 1 / x ) n 1 ,
then since q n = ( 10 n + 1 1 ) / 9, your number is
n = 1 1 q n = 9 f ( 1 10 ) 1.
Chowla proved that f ( 1 10 ) is irrational, and hence so is your number.

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