How can understanding limits help solve infinite series in Calculus II?

[MillerGenuine]

Homework Statement

$$\sum_{n=1}^\infty(-1)^n \frac{n^n}{n!}$$

Homework Equations

I can not find my limit as n approaches infinity. I know that the answer is infinity but I am not sure how to get it.

The Attempt at a Solution

 

[╔(σ_σ)╝]

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

What can you say about

$$<br /> \lim_{n \rightarrow \infty} \frac{n^n}{n!}<br />$$

Does it go to zero ?

EDIT
Fixed

 

[MillerGenuine]

Seems to me that it would become $$\frac{\infty}{infty}=1$$

 

[╔(σ_σ)╝]

Seems to me that it would become $$\frac{\infty}{infty}=1$$

How did you arrive at that answer ?

It is not correct.

$$\lim_{n \rightarrow \infty} \frac{n^n}{n!} = \lim_{n \rightarrow \infty} \frac{n*n*n*...*n}{1*2*3...*n} = \lim_{n \rightarrow \infty} \frac{n*n*n*...*n}{1*2*3...*n-1}<br /> <br />$$
As n gets very large

n/1 = n
n/2 > 1
n/3 > 1
...
n/n-1 $$\cong 1$$

 

[MillerGenuine]

I just based it on the fact that i have an "n" in the numerator and an "n" in the denominator so plugging in infinity for both will give me 1. Which is clearly incorrect, but i just have no idea how to approach this problem.

 

[╔(σ_σ)╝]

I just based it on the fact that i have an "n" in the numerator and an "n" in the denominator so plugging in infinity for both will give me 1. Which is clearly incorrect, but i just have no idea how to approach this problem.

Well look at what I gave you .

Or simply look at a pattern

Lets use some examples

n=1

1^1/ 1! =1

2^2 /2! = 4/2 =2

3^3 /3! = 9/2



Can you see that n^n /n! is increasing and not bounded above.

In fact

n^n/n! > = n

 

[MillerGenuine]

Im sorry but its just not clicking for me. I see what you have shown and all i keep seeing in my head is infinity^infinity/infinity! which seems to be infinity/infinity=1
my apoogies if this sounds stupid but bare with me

 

[╔(σ_σ)╝]

Im sorry but its just not clicking for me. I see what you have shown and all i keep seeing in my head is infinity^infinity/infinity! which seems to be infinity/infinity=1
my apoogies if this sounds stupid but bare with me

Don't plug in infinity like that. Your thinking is dangerous

If we followed your logic then we could prove that
$$\lim_{n \rightarrow \infty} \frac{n}{n^2 +1} = 1$$

Read what I posted carefully maybe something would click or go to bed and attempt the problem tomorrow ( assuming it's night where you are it's 1:27 am here ).

 

[Mark44]

Seems to me that it would become $$\frac{\infty}{\infty}=1$$

$$\frac{\infty}{\infty}$$ is not a number. This is one of several indeterminate forms, others of which include 0/0, $$\infty - \infty, and$$, $$1^{\infty}$$

These are all symbolic and do not represent numbers. When you have a limit expression whose limit is any of these, the actual limit can come out to be anything.

Going back to what ╔(σ_σ)╝ said, a_n for your series is nⁿ/n!. This is
$$\frac{n \cdot n \cdot n \cdot ... \cdot n}{1 \cdot 2 \cdot 3 \cdot ... \cdot n}$$

or
$$\frac{n}{1} \cdot \frac{n}{2} \cdot \frac{n}{3} \cdot ... \cdot \frac{n}{n}$$

As n gets large, the last factor on the right remains 1, but what is happening to the factors on the left?

 

[MillerGenuine]

Read what I posted carefully maybe something would click or go to bed and attempt the problem tomorrow

I think i may take your advice on going to bed & take a fresh look at it tomorrow because i just can not seem to understand this. I am sure if either of you were to explain in person It would be much easier to understand. I think the main problem is my lack of understanding the concepts of this class. my prof only teaches mechanics of problems so i struggle with conceptual calculus

 

[╔(σ_σ)╝]

I think i may take your advice on going to bed & take a fresh look at it tomorrow because i just can not seem to understand this. I am sure if either of you were to explain in person It would be much easier to understand. I think the main problem is my lack of understanding the concepts of this class. my prof only teaches mechanics of problems so i struggle with conceptual calculus

It's okay; my brain also I not functioning as I would like at this point too :).

Take some time to review limits as this is indispensable to discussions on infinite series.