Why is lim_(n->infty) ((n-1)/(n+1))^(2n+4)=1/e^4

Jefferson Booth

Jefferson Booth

Answered question

2022-11-02

Why is lim n ( n 1 n + 1 ) 2 n + 4 = 1 e 4

Answer & Explanation

Gilbert Petty

Gilbert Petty

Beginner2022-11-03Added 23 answers

If you already know that
lim n ( 1 + x f ( n ) ) f ( n ) = e x , x R
and for any function f s.t. f ( n ) n , then
( n 1 n + 1 ) 2 n + 4 = [ ( 1 2 n + 1 ) n + 1 ] 2 ( 1 2 n + 1 ) 2 n ( e 2 ) 2 1 2 = e 4
Alfredo Cooley

Alfredo Cooley

Beginner2022-11-04Added 4 answers

Another way to solve:
lim n ( n 1 n + 1 ) 2 n + 4
= lim n ( ( 1 + ( 2 ) n + 1 ) n + 1 2 ) 2 ( 2 n + 1 ) n + 1
= ( lim n ( 1 + ( 2 ) n + 1 ) n + 1 2 ) lim n ( 4 2 n 1 + 1 n )
= ( e ) 4  as  n , n + 1 2  and  lim m ( 1 + 1 m ) m = e

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