I want to find the limit of the following sequence x n </msub> = (

Hailie Blevins

Hailie Blevins

Answered question

2022-06-11

I want to find the limit of the following sequence
x n = ( log ( n + 2 ) log ( n + 1 ) ) n log n

Answer & Explanation

plodno8n

plodno8n

Beginner2022-06-12Added 17 answers

The logarithmic equivalence criterion which you are referring seems not necessary, let use instead
( log ( n + 2 ) log ( n + 1 ) ) n log n = e n log n log ( log ( n + 2 ) log ( n + 1 ) )
and by
log ( n + 2 ) log ( n + 1 ) = 1 + log n + 2 n + 1 log ( n + 1 )
log ( n + 2 ) log ( n + 1 ) = 1 + log n + 2 n + 1 log ( n + 1 )
we have
n log n log ( log ( n + 2 ) log ( n + 1 ) ) = log ( 1 + log n + 2 n + 1 log ( n + 1 ) ) log n + 2 n + 1 log ( n + 1 ) n log n log n + 2 n + 1 log ( n + 1 )
and
n log n log n + 2 n + 1 log ( n + 1 ) = log n log ( n + 1 ) log ( 1 + 1 n + 1 ) n 1 log e = 1
then
( log ( n + 2 ) log ( n + 1 ) ) n log n = e n log n log ( log ( n + 2 ) log ( n + 1 ) ) e 1 = e

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

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

Didn't find what you were looking for?