Does Sequence (n,1/n) Converge or Diverge?

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So in summary, the sequence (n, 1/n) converges in the x-y plane, but only when considering the points (1, 1/n). When looking at the points (n, 1/n), the sequence diverges. This is because for a sequence to converge in R^2, both x and y values must converge separately, and in this case, only the y values converge while the x values diverge. This concept is important to understand when studying sequences and their convergence.
  • #1
zendani
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if we have a sequence (n,1/n) , n E N , the sequence converges?

lim n = infinite
lim 1/n = 0

(1,1),(2,1/2),(3,1/3)...(n,1/n)

it is convergent and divergent?!
 
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  • #2
if anybody knows about such a sequence, book or reference, please write here

because i want to learn it

Thank you
 
  • #3
zendani said:
if anybody knows about such a sequence, book or reference, please write here

because i want to learn it

Thank you

In order to converge in R^2, the x-y plane, a sequence of points has to converge in each variable separately. So the sequence (1, 1/n) does not converge.
 
  • #4
For a sequence of the form (xn,yn) to converge, we require that both xn and yn converges. Here, xn=n, yn=1/n. While yn converges to 0, xn diverges so we say that (n,1/n) diverges.
 
  • #5
thank you Stevel27 and quasar987, i got it

stevel, i have (n,1/n) no (1,1/n)

so (n, 1/n) diverges and (1,1/n) converges...
 
  • #6
correct! :)
 
  • #7
zendani said:
thank you Stevel27 and quasar987, i got it

stevel, i have (n,1/n) no (1,1/n)

so (n, 1/n) diverges and (1,1/n) converges...

Yes, you're right about that. Typo on my part, but of course (1, 1/n) does converge.
 

1. What is the sequence (n,1/n)?

The sequence (n,1/n) is a mathematical sequence where the first term is n and the second term is 1/n. The following terms are n+1 and 1/(n+1), and so on.

2. What does it mean for a sequence to converge or diverge?

A sequence converges if its terms approach a specific value as n increases. This value is called the limit of the sequence. A sequence diverges if its terms do not approach a specific value as n increases.

3. Does the sequence (n,1/n) converge or diverge?

The sequence (n,1/n) diverges. As n increases, the terms alternate between getting closer to 0 and getting closer to infinity. Therefore, there is no specific value that the terms approach, indicating divergence.

4. How can you determine if a sequence converges or diverges?

There are several methods for determining the convergence or divergence of a sequence, such as the limit comparison test, ratio test, and root test. In the case of the sequence (n,1/n), we can also observe the behavior of the terms and see that they do not approach a specific value.

5. What are some real-world applications of the sequence (n,1/n)?

The sequence (n,1/n) has applications in mathematics, physics, and engineering. For example, it can be used to model the behavior of electrical circuits and the decay of radioactive substances. It is also used in the study of series and sequences in calculus.

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