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

Holetaug

Holetaug

Answered question

2022-07-14

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

Answer & Explanation

Jaelynn Cuevas

Jaelynn Cuevas

Beginner2022-07-15Added 16 answers

We can use the fact that if a n > 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 .

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