Question

Help me with finding the limit of this sequence again, I have to use the squeeze theorem here.

Answer 1

\[\lim_{n \rightarrow \infty} n ^{n}/n!\]

Answer 2

It would seem that the limit is infinity, right?

Answer 3

So to use squeeze theorem, we just need something which is obviously less than that function, yet approaches infinity.

Answer 4

Can u please elaborate?

Answer 5

On why I think it approaches infinity?

Answer 6

The basic intuition is this that it approaches to infinity but to use squeeze theorem I need its bounds which I have not been able to figure it out

Answer 7

n^n = n*n*n*n*n (n times)
n! = 1*2*3*...*n

So n^n/n! can be written as n/1 * n/2 * n/3*...*n/n

Which is n multiplied by a number of things which are all greater than or equal to 1. Since n goes to infinity, the product must go to infinity.

Answer 8

Oh I know.

Answer 9

Just use n. n is always less than the sequence, but approaches infinity.

Answer 10

And an upper bound that works is n^n, which obviously approaches infinity.

Answer 11

BAM. SANDWHICHED.

Answer 12

And which is the lower bound here

Answer 13

n is the lower bound. Do you see why? I rewrote the function as n/1*n/2*n/3*...*n/n which is just n multiplied by a bunch of other things which are all greater than or equal to 1. So the function is greater than n.

Answer 14

n < n^n/n!

Answer 15

lim n->infinity = infinity

Answer 16

The upper bound is kind of extraneous after that, but n^n works well.

Answer 17

Why n^n is the upper bound here?

Answer 18

Because the sequence is n^n / n!
n! is greater than one, so obviously if you don't divide by n!, then the sequence will be greater.

Answer 19

Did I explain the lower bound part well enough? Did that make sense?

Answer 20

I am still kinda confused with the lower bound here

Answer 21

http://www.twiddla.com/589333

Answer 22

I am multiplying n by the numbers equal or less than n so ..........

Answer 23

I don't have microphone here so I will be in touch with u only through chat, will that be fine?