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Question about series and sequences

  • #1
188
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?
 

Answers and Replies

  • #2
180
4
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
188
0
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
180
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 text book/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:

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