Why the limit of the following sequence is like this: 2 n ! n n = 2 e

Question

Why the limit of the following sequence is like this:            2                        n          !                n              n    =      2    e

Jaelynn Cuevas · Accepted Answer

We can use the fact that if       a    n    &amp;gt;  0 for all   n  ≥  1 and the sequence             a              n        +        1                    a      n       converges in   [  0  ,  ∞  ], then      lim          n      →      ∞            a    n          1              /            n        =      lim          n      →      ∞                  a              n        +        1                    a      n      We have that       a    n    =  n  !      /        n    n   and      lim          n      →      ∞                  a              n        +        1                    a      n        =      lim          n      →      ∞                  (      n      +      1      )      !              n        n                    (      n      +      1              )                  n          +          1                    n      !        =      lim          n      →      ∞                  n      n              (      n      +      1              )        n              =      1                  lim                  n          →          ∞                    (      1      +      1              /            n              )        n              =      1    e    .

Why the limit of the following sequence is like this: 2 <mroot>

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