Need help proving convergence of this series

In summary, the conversation discusses the convergence of a sequence and series, and explores different methods such as the ratio test and Stirling's approximation to determine the behavior and convergence of the sequence. The conclusion is that the series converges due to the monotone convergence theorem and the comparison test with the convergent series of x_n = 1/n^2.
  • #1
swampwiz
571
83
n! / nn

I have proven that the sequence converges numerically, but I can't do it analytically, and can't do anything for the series (maybe it the series doesn't converge?)
 
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  • #2
Can you find a relation between the n-th term and the n+1-th term?

Or if you want the asymptotic behavior you can use the stirling's approximation
 
  • #3
Use the ratio test
 
  • #4
First note its a decreasing sequence bounded by 0, so by the monotone convergence thereom it converges to some number.

If the series is going to converge, the sequence needs to go to 0, which it does as n!/n^n <= 1/n^2 (so use the squeze theorem).

By the p test(you could also use ratio test), we know the series where x_n = 1/n^2 converges.

Thus by the comparision test we see that the series with x_n = n!/n^n converges.
 

FAQ: Need help proving convergence of this series

1. What is a series?

A series is a mathematical expression that represents the sum of a sequence of numbers, typically written as a1 + a2 + a3 + .... It is important to note that the terms in a series must follow a specific pattern and cannot be random numbers.

2. What does it mean for a series to converge?

A series converges if the sum of its terms approaches a finite value as more terms are added. In other words, as the number of terms in the series increases, the sum of those terms becomes closer and closer to a specific number.

3. How do you prove convergence of a series?

To prove convergence of a series, you must show that the limit of the sum of its terms approaches a finite value as the number of terms approaches infinity. This can be done using various techniques, such as the comparison test, the ratio test, or the integral test.

4. What is the difference between absolute convergence and conditional convergence?

A series is absolutely convergent if the sum of the absolute values of its terms converges. On the other hand, a series is conditionally convergent if the sum of its terms converges, but the sum of the absolute values of its terms diverges.

5. Can a series diverge even if its terms approach zero?

Yes, it is possible for a series to diverge even if its terms approach zero. This can happen if the terms do not decrease fast enough to offset the increase in the number of terms being added. In such cases, other tests, such as the alternating series test, may be used to determine convergence.

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