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?

Question

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?

Aaron Everett · Accepted Answer

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      !        &amp;lt;      1          (      q      +      1      )      (      q      +      2      )        ⋅      ∑          m      =      0            1          2      m        =      2          (      q      +      1      )      (      q      +      2      )        ,which implies that  0  &amp;lt;      1          q      +      1        −      2          (      q      +      1      )      (      q      +      2      )        ≤      |    q  !  ⋅      ∑          n      =      q      +      1        ∞              ϵ      n              n      !            |    ≤      1          q      +      1        +      2          (      q      +      1      )      (      q      +      2      )        &amp;lt;  1  ,a contradiction.

Let { ϵ n } be a sequence where ϵ n is either 1 or...

Answered question

Answer & Explanation