I need to solve the following problem: <munder> <mo movablelimits="true" form="prefix">lim

Summer Bradford

Summer Bradford

Answered question

2022-06-27

I need to solve the following problem:
lim n ( n n 1 ) 1 n .

Answer & Explanation

Kaydence Washington

Kaydence Washington

Beginner2022-06-28Added 32 answers

Let n be a positive integer.
Using Landau notations, n n 1 = exp ( 1 n ln ( n ) ) 1 = ln ( n ) n + o ( ln ( n ) n ) as n +
Then 1 n ln ( n n 1 ) = 1 n ln ( ln ( n ) n + o ( ln ( n ) n ) ) 0 when n +
And ( n n 1 ) 1 n = exp ( 1 n ln ( n n 1 ) ) exp ( 0 ) = 1 when n +
Celia Lucas

Celia Lucas

Beginner2022-06-29Added 5 answers

We have for n>2
ln ( n n 1 ) n = ln [ n n ( 1 1 n n ) ] n
= ln ( n n ) n + ln ( 1 1 n n ) n
= ln ( n ) n 2 + ln ( 1 1 n n ) n
But for x>1, ln ( x ) x 1 so
0 ln ( n ) n 2 n 1 n 2
and the sequence ( ln ( 1 1 n n ) ) n 2 is bounded

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