Limit with Factorial: Understanding 3/(n+1) Limit

In summary, the conversation discusses simplifying the expression [3^(n+1)]/(n+1)! * [n!/(3^n)] and how to approach it by using algebraic properties. It is concluded that the expression can be simplified to 3/(n+1).
  • #1
Kuno
19
0

Homework Statement


Why does the limit as n -> infinity of [3^(n+1)]/(n+1)!] * n!/(3^n) equal
the limit as n -> infinity of 3/(n+1)?

Homework Equations


The Attempt at a Solution


I have never encountered this before.
 
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  • #2
Forget about the limit and just focus on simplifying algebraically. You know that [tex]{(n+1)}! = n!{(n+1)}[/tex]. Also [tex]3^{n+1} = (3)(3^n)[/tex]. Use those to cancel some terms and see what you get.
 
  • #3
Forget about the limit, how do you simplify:

[3^(n+1)]/(n+1)! * [n!/(3^n)]

(LOL, Curious3141 is faster than me)
 
  • #4
yenchin said:
Forget about the limit, how do you simplify:

[3^(n+1)]/(n+1)! * [n!/(3^n)]

(LOL, Curious3141 is faster than me)

And it's weird how we worded that almost identically! :eek:
 
  • #5
Ah yes, that makes perfect sense. Thanks.
 

1. What is the limit of 3/(n+1) as n approaches infinity?

The limit of 3/(n+1) as n approaches infinity is 0. This is because as n gets larger and larger, the denominator (n+1) becomes significantly larger than the numerator (3), making the fraction approach 0.

2. How do you calculate the limit of 3/(n+1)?

To calculate the limit of 3/(n+1), you can use the limit laws and algebraic manipulation to rewrite the expression into a form where you can easily see the limit. In this case, you can divide both the numerator and denominator by n, which will give you the expression 3/n. Then, you can apply the limit law that states the limit of a constant (in this case, 3) is equal to the constant. Therefore, the limit of 3/(n+1) is equal to 3/n, which approaches 0 as n approaches infinity.

3. Is the limit of 3/(n+1) a finite value?

Yes, the limit of 3/(n+1) is a finite value. As n approaches infinity, the fraction approaches 0, which is a finite number. This means that the limit of 3/(n+1) exists and is equal to 0.

4. Can you use L'Hopital's rule to find the limit of 3/(n+1)?

No, L'Hopital's rule cannot be used to find the limit of 3/(n+1). L'Hopital's rule can only be used for limits involving indeterminate forms such as 0/0 or infinity/infinity. In this case, the limit of 3/(n+1) is not an indeterminate form, so L'Hopital's rule cannot be applied.

5. How does the factorial affect the limit of 3/(n+1)?

The factorial does not affect the limit of 3/(n+1). As n approaches infinity, the factorial term becomes insignificant compared to the n term, so it does not affect the overall limit. This is because factorial grows much faster than a polynomial function (such as n), so it becomes negligible in the limit calculation.

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