A problem with the convergence of a series

• Amaelle
In summary, the conversation discusses the convergence of a series using the Racine test. The test shows that the series converges to 2/3. However, the speaker is confused because for a sequence to be convergent, the term an should go to 0, but the limit of the series is infinity. They are then advised to reconsider their approach and check for any mistakes, which leads to them realizing that the denominator is also inside the logarithm. The conversation ends with the speaker thanking the other person for their help.
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!

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
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.

Amaelle
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!

vela
FactChecker said:
The denominator, 3n+1, is also inside the log.
thank you it's clear now

1. What is a series convergence problem?

A series convergence problem refers to a situation where a mathematical series, which is an infinite sum of numbers, does not have a finite limit or does not converge to a specific value. This can occur due to various reasons, such as the series having infinitely many terms, having terms that do not approach zero, or having alternating signs.

2. How do you determine if a series converges or diverges?

To determine if a series converges or diverges, we can use various tests such as the comparison test, ratio test, or integral test. These tests involve comparing the given series to a known convergent or divergent series and using mathematical techniques to determine the behavior of the given series.

3. What is the importance of convergence in series?

Convergence in series is important because it ensures that the sum of the terms in the series has a finite value. This is crucial in many fields of mathematics and science, as it allows for the use of series to approximate functions and solve problems. In addition, convergence also helps in determining the behavior and properties of a series.

4. Can a series converge to more than one value?

No, a series can only converge to one value. This is because convergence implies that the sum of the series approaches a specific value as the number of terms increases. If a series were to converge to more than one value, it would not have a unique limit, which goes against the definition of convergence.

5. What are some common causes of series convergence problems?

Some common causes of series convergence problems include the presence of infinitely many terms, terms that do not approach zero, or terms that alternate in sign. In addition, series with rapidly increasing or decreasing terms can also lead to convergence problems. It is important to carefully analyze the properties of a series to identify the cause of the convergence problem.

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