A problem with the convergence of a series

[Amaelle]

Homework Statement

Show that the following sequence is convergent

Relevant Equations

Racine test

Good day
I have a question about the convergence of the following serie

I understand that the racine test shows that it an goes to 2/3 which makes it convergent
but I also know that for a sequence to be convergent the term an should goes to 0 but the lim(n---->inf) ((2n+100)/(3n+1))^n)=lim exp(n*log(2n+100)/(3n+1))=+infinity
I'm really confused
thank you!

 

[fresh_42]

What do you get if you cancel the quotient by $n$ instead of taking the logarithm?

 

[vela]

lim exp(n*log(2n+100)/(3n+1))=+infinity

You should rethink this part. Try plugging in a large value of $n$ and see if it's what you expect.

 

[FactChecker]

Homework Statement:: Show that the following sequence is convergent
Relevant Equations:: Racine test

Good day
I have a question about the convergence of the following serie
View attachment 283529

I understand that the racine test shows that it an goes to 2/3 which makes it convergent
but I also know that for a sequence to be convergent the term an should goes to 0 but the lim(n---->inf) ((2n+100)/(3n+1))^n)=lim exp(n*log(2n+100)/(3n+1))=+infinity
I'm really confused
thank you!

The denominator, 3n+1, is also inside the log.

 

[Amaelle]

You should rethink this part. Try plugging in a large value of $n$ and see if it's what you expect.

thank you very much I just spotted the mistake!

 

[Amaelle]

The denominator, 3n+1, is also inside the log.

thank you it's clear now