Proving convergence of factorial w/o Ratio Test

In summary, the conversation discusses determining whether 1/n! diverges or converges using different tests such as comparison tests, p-series, geometric series, divergence test, and integral test. The conversation also mentions using limit comparison with b_n=1/n^n, but the speaker is unsure if it converges or not. Another suggestion is to use the Comparison Test with the series n!>n^2 for sufficiently large n.
  • #1
mvpshaq32
28
0

Homework Statement


Determine whether 1/n! diverges or converges.
So far, we have only learned the comparison tests, p-series, geometric series, divergence test, and integral test, so I can only use these tests to prove it.


Homework Equations



N/a

The Attempt at a Solution



I thought about using limit comparison with my b_n=1/n^n, but I can't determine if that converges or not, so I don't know what to do.
 
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  • #2
Why did you choose [itex]b_n = 1/n^n[/itex]? Why not a simpler series that you know converges or diverges?
 
  • #3
The Comparison Test is your friend
 
  • #4
It should be clear that for n> 3, n^2> n!.
 
  • #5
HallsofIvy said:
It should be clear that for n> 3, n^2> n!.

I think you meant 2^n < n! ? Easy typo to make.

Oops, it's 2^(n-1) < n!
 
  • #6
It's monotonic, so if you show that it is bounded, you're in business. Now by the comparison test, we have n!>n^2 for sufficiently large n, so take reciprocals and go from there.
 

FAQ: Proving convergence of factorial w/o Ratio Test

1. What is the purpose of proving convergence of factorial without using the Ratio Test?

The purpose of this proof is to determine the convergence of a factorial series without relying on the Ratio Test, which may be difficult or impossible to use in certain cases.

2. How is convergence of factorial without using the Ratio Test typically proven?

Convergence of factorial can be proven using other methods such as the Comparison Test, the Root Test, or the Integral Test.

3. What are the limitations of using the Ratio Test to prove convergence of factorial?

The Ratio Test may not work for all factorial series, especially those with alternating signs or with terms that do not tend to zero. It also requires the limit of the ratio of consecutive terms to be taken, which can be time-consuming or challenging to calculate in some cases.

4. What are the benefits of proving convergence of factorial without using the Ratio Test?

By using alternative methods to prove convergence, we can gain a better understanding of the behavior of the factorial series and potentially find simpler or more efficient ways to prove convergence.

5. Can the methods used to prove convergence of factorial without using the Ratio Test be applied to other types of series?

Yes, the Comparison Test, Root Test, and Integral Test can be applied to other types of series as well to determine their convergence. However, the specific techniques used may vary depending on the type of series being analyzed.

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