Yes, I meant c>0. Thank you for clarifying!

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    Divergence
In summary, the conversation discusses the convergence of the sequence 2^(n^2)/n!, with one person suggesting to show that 2^(n^2)/n^n diverges by looking at the logarithm. The other person confirms that if lim a_n = infinity and |b_n|< c < infinity, then lim |a_n*b_n| = infinity, as long as c>0.
  • #1
peripatein
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Hi,
How may I show that 2^(n^2)/n! converges to infinity?
 
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  • #2
peripatein said:
Hi,
How may I show that 2^(n^2)/n! converges to infinity?

That is SO divergent you can afford to be pretty sloppy. 2^(n^2)/n!>2^(n^2)/n^n, right? So show 2^(n^2)/n^n diverges. Hint: look at the log.
 
  • #3
Thank you, Dick!
 
  • #4
Would it be correct to say that if for sequences a_n and b_n, lim a_n = infinity and |b_n|< c < infinity, then lim|a_n*b_n| = infinity?
(I think it should be correct, as we may infer that lim |bn| = c and then the limit of the product of a_n and b_n would yield c*infinity which is always infinity.)
 
  • #5
peripatein said:
Would it be correct to say that if for sequences a_n and b_n, lim a_n = infinity and |b_n|< c < infinity, then lim|a_n*b_n| = infinity?
(I think it should be correct, as we may infer that lim |bn| = c and then the limit of the product of a_n and b_n would yield c*infinity which is always infinity.)

If you mean lim |b_n|=c with c>0, then sure. If c=0, then you need to think more.
 
Last edited:

1. What is the concept behind the divergence of 2^(n^2)/n?

The divergence of 2^(n^2)/n refers to the behavior of the sequence as n approaches infinity. In this case, the sequence has an exponential term in the numerator and a polynomial term in the denominator. As n gets larger and larger, the exponential term dominates and the sequence grows without bound, resulting in divergence.

2. How can the divergence of 2^(n^2)/n be mathematically proven?

Mathematically, we can prove that the sequence diverges by using the limit comparison test. By comparing the given sequence with the divergent geometric series 2^n, we can show that the limit of the ratio between the two sequences is equal to infinity as n approaches infinity, proving divergence.

3. Can the divergence of 2^(n^2)/n be visualized?

Yes, the divergence of this sequence can be visualized by plotting the values of the sequence for increasing values of n. As n gets larger, the graph will show a steep upward curve, indicating unbounded growth and thus, divergence.

4. Are there any real-world applications of the divergence of 2^(n^2)/n?

This concept has applications in various fields such as physics, engineering, and computer science. For example, it can be used to model the growth of populations, the spread of diseases, and the behavior of algorithms with increasing input size.

5. Are there any other similar sequences that exhibit divergence?

Yes, there are many other sequences that exhibit divergence, such as 2^n, n!, and n^n. These sequences have different combinations of exponential and polynomial terms, but they all share the common characteristic of unbounded growth as n approaches infinity.

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