Question about series and sequences

In summary, the function a_{n}=\frac{2+3n}{2n+1} is being analyzed to determine if it converges or diverges. The ratio test was used to find that the function diverges, and the question arises if the series (summation of a_n) also diverges. The expert explains that the series clearly converges, but the limit of the function is not 0, indicating that it diverges. The expert also clarifies that a function/sequence diverges if the limit is infinite and converges otherwise, and there is a possibility for the limit to not exist, as in the case of alternating sequences.
  • #1
warfreak131
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0

Homework Statement



I have a function [tex]a_{n}=\frac{2+3n}{2n+1}[/tex], and I have to find out whether it converges or diverges. I did the ratio test [tex]lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|[/tex]. And according to the divergence test, it should diverge. Then it asks if the series, (summation of [tex]a_n[/tex]) converges or diverges.

Now my question is, if a sequence diverges, does the series converge as well?
 
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  • #2
Why do you say the sequence diverges?

[tex]a_{n}=\frac{2+3n}{2n+1} =\frac{3}{2} + \frac{1}{4n+2}[/tex]

Clearly converges. As to the series, you can see that every term is at least 3/2.
 
  • #3
The divergence test says that if the limit of a function is not 0, then it diverges. And the limit is 3/2, which is not 0, so it diverges... right?
 
  • #4
I was under the impression that a function/sequence diverges if the limit is infinite and converges otherwise, though admittedly it's been some time since I had anything to do with this. I'm sure you have a definition in your textbook/lecture notes. Are you sure you're not thinking of the divergence of a series? A series does indeed diverge if the elements don't approach 0.

edit: there is also a possibility that the limit doesn't exit, like with alternating sequences, in which case the sequence diverges also.
 
Last edited:

Related to Question about series and sequences

1. What is a series in mathematics?

A series is a sum of numbers, with each number in the sum called a term. It can be represented using the sigma notation as Σan, where n is the index of the terms and a is the general term or rule that defines the terms of the series.

2. What is a sequence in mathematics?

A sequence is a list of numbers that follow a specific pattern or rule. It can be represented using the subscript notation as a1, a2, a3, ... , an, where a is the general term or rule and n is the index of the terms in the sequence.

3. What is the difference between a series and a sequence?

The main difference between a series and a sequence is that a series is a sum of numbers, while a sequence is a list of numbers. A series has a finite or infinite number of terms, while a sequence has a finite number of terms.

4. How do you find the sum of a series?

To find the sum of a series, you can use various methods such as the partial sum method, the telescoping series method, or the ratio test. These methods involve finding the general term of the series and using mathematical operations to evaluate the sum.

5. What are some real-life applications of series and sequences?

Series and sequences have many real-life applications in fields such as physics, engineering, and finance. For example, in physics, series are used to calculate the distance and displacement of objects, while sequences are used to model motion and predict future positions. In finance, series and sequences are used to calculate interest rates and investment growth.

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