a_n=(1+1/n)^n: Prove the following Equation. We have to show that (a_(n+1))/(a_n) = (1-1/((n+1)^2))^(n+1) (n+1)/(n) , n in NN I would really appreciate any help and I hope everything is correctly written.

videosfapaturqz

videosfapaturqz

Answered question

2022-09-24

a n = ( 1 + 1 n ) n : Prove the following Equation.
I have trouble solving following problem:
Given the sequence:
a n = ( 1 + 1 n ) n
We have to show that
a n + 1 a n = ( 1 1 ( n + 1 ) 2 ) n + 1 n + 1 n , n N
I would really appreciate any help and I hope everything is correctly written.
Greetings

Answer & Explanation

ahem37

ahem37

Beginner2022-09-25Added 15 answers

Because
a n + 1 a n = ( n + 2 ) n + 1 ( n + 1 ) n + 1 ( n + 1 ) n n n = n + 1 n ( n + 2 ) n + 1 n n + 1 ( n + 1 ) 2 n + 2 =
= n + 1 n ( n 2 + 2 n ( n + 1 ) 2 ) n + 1 = ( 1 1 ( n + 1 ) 2 ) n + 1 n + 1 n .

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