How to show this series is divergent?

In summary, to identify if a series is divergent, you can use the divergence test or the nth term test, which states that if the limit of the nth term of the series as n approaches infinity is not equal to zero, then the series is divergent. The comparison test can also be used to show a series is divergent by comparing it to a known divergent series. Other tests such as the integral test, ratio test, and root test can also be used depending on the form of the series. A series cannot be both convergent and divergent. Proving a series is divergent is significant as it helps us understand the behavior of the series and its sum, which can be useful in various applications.
  • #1
KFC
488
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I got a series in the following form

[tex]\sum_{k=0}^\infty\dfrac{k!}{c^{2(k-1)}}[/tex]

where c is arbritary complex constant, how can I show this series is divergent?

Thanks
 
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  • #2
You have a factorial on top. It looks fairly obvious to me.

If it's not obvious to you, try a ratio test.
 
  • #3
Yep, ratio test.
 

1. How do I identify if a series is divergent?

To determine if a series is divergent, you can use the divergence test, also known as the nth term test. This test states that if the limit of the nth term of the series as n approaches infinity is not equal to zero, then the series is divergent.

2. Can I use the comparison test to show a series is divergent?

Yes, the comparison test can be used to show a series is divergent. This test states that if a series is greater than or equal to another series that is known to be divergent, then the original series must also be divergent.

3. Are there any other tests I can use to prove a series is divergent?

Yes, there are several other tests that can be used to show a series is divergent. These include the integral test, the ratio test, and the root test. It is important to choose the appropriate test based on the form of the series.

4. Can a series be both convergent and divergent?

No, a series cannot be both convergent and divergent. By definition, a series is either convergent or divergent, but not both. If a series is divergent, it means that the terms of the series do not approach a finite limit as the number of terms increases.

5. What is the significance of proving a series is divergent?

Proving that a series is divergent is important because it helps us understand the behavior of the series and whether it has a finite sum or not. If a series is divergent, it means that the sum of an infinite number of terms is either infinite or does not exist. This information can be useful in various applications, such as in calculating infinite sums or in analyzing the convergence of numerical methods.

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