A problem with the convergence of a series

Amaelle
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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
inf.png


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!
 
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What do you get if you cancel the quotient by ##n## instead of taking the logarithm?
 
Amaelle said:
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.
 
Amaelle said:
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.
 
vela said:
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!
 
FactChecker said:
The denominator, 3n+1, is also inside the log.
thank you it's clear now
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...

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