Determining Divergence or Convergence in Series

In summary, the conversation discusses using the Ratio Test and Limit Comparison Test to determine the convergence or divergence of the series \sum (2n^{2}+3n)/\sqrt{5+n^{5}} with an index of n=1 to infinity. The individual pieces of the series are suggested to be evaluated separately to determine the convergence of the entire series.
  • #1
badirishluck
6
0

Homework Statement


[tex]\sum[/tex] (2n[tex]^{2}[/tex]+3n)[tex]/\sqrt{5+n^{5}}[/tex]
index n=1 to infinity

Homework Equations





The Attempt at a Solution


I tried both the Ratio Test (limit as n goes to infinity of a[tex]_{n+1}[/tex][tex]/a_{n}[/tex]) and the Limit comparison test (limit as n goes to infinity of [tex]a_{n}[/tex]/ [tex]b_{n}[/tex]) but wasn't able to come up with the same answer from the two tests. What am I doing wrong?
Does it converge or diverge? How?
 
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  • #2
try breaking it into 2 pieces and see what you can do with the individual pieces, see if
[tex]\sum[/tex] (2n[tex]^{2}[/tex])[tex]/\sqrt{5+n^{5}}[/tex]
converges, if it does then the whole thing does since the 2nd term is smaller, if it doesn't then the whole thing does not converge.
 

Related to Determining Divergence or Convergence in Series

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

Divergence in series means that the terms of the series tend to infinity as the number of terms increases, while convergence means that the terms tend to a finite limit as the number of terms increases.

2. How do you determine if a series is convergent or divergent?

There are several tests that can be used to determine convergence or divergence in series, such as the comparison test, the ratio test, and the integral test. These tests involve analyzing the behavior of the terms in the series to determine if they tend to a finite limit or infinity.

3. Can a series be both convergent and divergent?

No, a series can only be either convergent or divergent. If a series is convergent, it cannot also be divergent, and vice versa.

4. What is the significance of determining convergence or divergence in series?

Determining convergence or divergence in series is important in mathematics and physics, as it allows us to understand the behavior of a series and make predictions about its behavior as the number of terms increases. This can be useful in solving real-world problems and making accurate calculations.

5. Are there any series that are always convergent or divergent?

Yes, there are certain series that are always convergent or divergent, regardless of the values of their terms. For example, the geometric series is always convergent if the absolute value of the common ratio is less than 1, and always divergent if the absolute value is greater than or equal to 1.

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