Prove that the following sequence is increasing e <mrow class="MJX-TeXAtom-ORD">

Mary Ashley

Mary Ashley

Answered question

2022-06-03

Prove that
the following sequence is increasing
e n = ( 1 + 1 n ) n , n 1 ;

Answer & Explanation

Joshua Nelson

Joshua Nelson

Beginner2022-06-04Added 1 answers

In order to prove that the given sequence is strictly increasing, we are to demonstrate e n + 1 > e n :
( 1 + 1 n + 1 ) n + 1 > ( 1 + 1 n ) n .
Let's rewrite the inequality above as:
( 1 + 1 n + 1 1 + 1 n ) n > 1 1 + 1 n + 1 .
The right-hand side equals
1 1 + 1 n + 1 = n + 1 n + 2 = 1 1 n + 2 .
Now, let's focus on the left-hand side:
( 1 + 1 n + 1 1 + 1 n ) n = ( ( n + 2 ) / ( n + 1 ) ( n + 1 ) / n ) n = ( n ( n + 2 ) ( n + 1 ) 2 ) n = ( 1 1 ( n + 1 ) 2 ) n .
By the Bernoulli's inequality, the following holds:
( 1 1 ( n + 1 ) 2 ) n 1 n ( n + 1 ) 2 Now it's purely technical to show the desired inequality
1 n ( n + 1 ) 2 > 1 1 n + 2 ,
because
n ( n + 1 ) 2 < 1 n + 2 .

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