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vrotterigzl

vrotterigzl

Answered question

2022-06-20

Let { ϵ n } be a sequence where ϵ n is either 1 or 1. How can We Showthat the sum of the series
n = 0 ϵ n n !
is an irrational number?

Answer & Explanation

Aaron Everett

Aaron Everett

Beginner2022-06-21Added 18 answers

If there exist p Z and q N , such that p q = n = 0 ϵ n n ! , then q ! n = q + 1 ϵ n n ! must be an integer. However,
| q ! n = q + 1 ϵ n n ! ϵ q + 1 q + 1 | n = q + 2 q ! n ! < 1 ( q + 1 ) ( q + 2 ) m = 0 1 2 m = 2 ( q + 1 ) ( q + 2 ) ,
which implies that
0 < 1 q + 1 2 ( q + 1 ) ( q + 2 ) | q ! n = q + 1 ϵ n n ! | 1 q + 1 + 2 ( q + 1 ) ( q + 2 ) < 1 ,
a contradiction.

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