Sequences & Series: Limit to Determine Convergence/Divergence

In summary, to determine if a sequence converges or diverges based on taking the limit, you can use the theorem that states if the sequence does not converge to 0, then the series cannot converge. If the sequence goes to infinity, it is considered to diverge.
  • #1
arctic_girl4
1
0
How do you know if a sequence converges or diverges based on taking the limit?

here's an example
f:= 3^n/n^3;

if i take the limit the sequence goes to infinity.

does it diverge becuase the limit is not zero or can the limit be something other than zero and it still converge?
 
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  • #2
You are talking about a sequences and not series? You titled this "sequences and series". There is a theorem that says that if a sequence does not converge to 0, then the series (infinite sum) of those numbers cannot converge but certainly if a sequence converges to any number then it converges!

However in this case the sequence "goes to" infinity, as you say, and infinity is not a number. The sequence diverges. Many textbooks would say this sequence "diverges to infinity".
 

Related to Sequences & Series: Limit to Determine Convergence/Divergence

1. What is the difference between convergence and divergence in sequences and series?

In sequences and series, convergence refers to the behavior of the terms in a sequence or the partial sums of a series approaching a specific value, known as the limit. Divergence, on the other hand, refers to the behavior of the terms or partial sums never reaching a specific value and instead becoming infinitely large.

2. How do you determine the convergence or divergence of a sequence or series?

To determine the convergence or divergence of a sequence or series, you can use various tests such as the limit comparison test, ratio test, and integral test. These tests involve taking the limit of the sequence or series and comparing it to known values to determine if it converges or diverges.

3. What is the role of the limit in determining the convergence or divergence of a sequence or series?

The limit plays a crucial role in determining the convergence or divergence of a sequence or series. It represents the value that the terms or partial sums are approaching, and if this value is finite, the sequence or series is convergent. If the limit is infinite or does not exist, the sequence or series is divergent.

4. Can a sequence or series be both convergent and divergent?

No, a sequence or series cannot be both convergent and divergent. By definition, if a sequence or series is convergent, it means the terms or partial sums are approaching a specific value, while divergence means they are not. Therefore, a sequence or series can only be one or the other.

5. Why is it important to determine the convergence or divergence of a sequence or series?

Determining the convergence or divergence of a sequence or series is essential because it tells us whether the terms or partial sums have a finite value or not. This information is crucial in many applications, such as calculating limits, approximating values, and understanding the behavior of functions.

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