Limit of n^2/n! and Using Limit Properties for Advanced Calculus Sequences

[MathSquareRoo]

Homework Statement

Determine whether the given limit exists and find their values. Give clear explanations using limit properties.

Homework Equations

lim n--->∞ (n^2)/n!

The Attempt at a Solution

I know that the limit is 0, but I don't know how to show it in detailed steps. Please help!

 

[jbunniii]

Well, you can immediately cancel an $n$ from the numerator and the denominator. Then try pairing the remaining $n$ in the numerator with one of the factors in the denominator, and see what you can conclude.

 

[Dick]

There are a lot of ways. Try one. Then someone can help. You have to TRY something. What are some ways you can show a sequence converges?

 

[MathSquareRoo]

I have already canceled the factor of n, and I am stuck at the next step. I have n/(n-1)(n-2)!

Any suggestions what to do next? How do I proof that =0?

 

[micromass]

This might be overkill for this problem, but try the squeeze theorem for a rigorous proof.

 

[jbunniii]

I have already canceled the factor of n, and I am stuck at the next step. I have n/(n-1)(n-2)!

Any suggestions what to do next? How do I proof that =0?

OK, so you have this:

$$\left(\frac{n}{n-1}\right)\left(\frac{1}{(n-2)!}\right)$$

Can you compute the limits of the two factors in parentheses?

 

[MathSquareRoo]

The limit of the first is 1, and the limit of the next is 0? Correct? Then can I simply multiply 1(0)=0. Will that be enough explanation?

 

[jbunniii]

The limit of the first is 1, and the limit of the next is 0? Correct? Then can I simply multiply 1(0)=0. Will that be enough explanation?

Yes, as long as you have the theorem that the limit of a product is the product of the limits. If not, you will either have to prove that, or find your limit a different way.

 

[MathSquareRoo]

Yes, I am able to use the product theorem. Thanks for the help!