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

Holetaug

Holetaug

Answered

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

Expert

2022-07-15Added 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 .

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?