Determine Limit of Sequence (n^n)/(n!) - Math Homework Help

In summary, the limit of a sequence is a number that the terms of the sequence approach as the index goes to infinity. To determine the limit, an expression for the general term must be found and various techniques can be used to simplify it. A convergent sequence approaches a finite limit, while a divergent sequence does not. Tests such as the ratio test and comparison test can be used to determine if a sequence is convergent or divergent. The limit of a sequence can be negative or imaginary, as it is simply the number that the terms approach.
  • #1
burritoloco
83
0

Homework Statement


Determine the limit of the sequence (n^n)/(n!)


Homework Equations





The Attempt at a Solution



I think the limit should be infinity as n^n grows faster than n!, but I'm not sure how to prove it. Thanks for the help!
 
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  • #2
Write it as

[tex]\frac n n\cdot \frac{n}{n-1}\cdot\frac{n}{n-2}\cdots\frac n 2\cdot\frac n 1[/tex]

and see if you can underestimate it with something going to infinity.
 
  • #3
Can you show that (n+1)(n+1)/(n+1)! > nn/n! for n>1 ?

This won't compete the proof, but it's a start.
 

Related to Determine Limit of Sequence (n^n)/(n!) - Math Homework Help

1. What is the definition of a limit of a sequence?

The limit of a sequence is a number that the terms of the sequence approach as the index (n) goes to infinity.

2. How do you determine the limit of a sequence?

To determine the limit of a sequence, you must first find an expression for the general term of the sequence. Then, you can use various techniques such as substitution, factoring, or algebraic manipulation to simplify the expression and find the limit as n approaches infinity.

3. What is the difference between a convergent and divergent sequence?

A convergent sequence is one in which the terms approach a finite limit as n goes to infinity. A divergent sequence is one in which the terms do not approach a finite limit, either because they increase or decrease without bound or because they oscillate between values.

4. How do you determine if a sequence is convergent or divergent?

To determine if a sequence is convergent or divergent, you can use various tests such as the ratio test, comparison test, or the limit comparison test. These tests analyze the behavior of the terms in the sequence and determine if they approach a finite limit or not.

5. Can the limit of a sequence be negative or imaginary?

Yes, the limit of a sequence can be negative or imaginary. The limit is simply the number that the terms of the sequence approach, regardless of its sign or whether it is a real or complex number.

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