Alternating Series involving factorials

[mateomy]

I have a specific problem but more than figuring out the answer I just want to figure out how to deal with factorials. My book is less than helpful on it...

The problem is...


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

I understand that I have to take the limit of the sequence (aside from determining an overall decreasing function), I just don't know what to do with the factorial. I am pretty sure I can't take a limit of a factorial. Can someone give me an example (using another problem if necessary) of what I should do with them.

Thanks.

 

[gb7nash]

Do you need to find out if the sum converges, or find the sum (assuming it converges)?

 

[mateomy]

I need to find convergence or divergence. Can I just pull out a factor of (1+n)! ? I saw an example like that in another text but the wording was slim to nil, it didn't really explain the circumstances when that could be done.

 

[Dick]

I need to find convergence or divergence. Can I just pull out a factor of (1+n)! ? I saw an example like that in another text but the wording was slim to nil, it didn't really explain the circumstances when that could be done.

When n=2, n^n/n! is 2*2/(1*2). When n=3, it's 3*3*3/(1*2*3). So the n=3 term is larger than the n=2 term. Looks to me like the sequence is INCREASING. Can you make an argument that it's increasing for all n? What would that say about convergence of the series? To get more practice with factorials, you could also try and apply a ratio test. What do you get for a_n+1/a_n? Can you say anything about it's limit?

 

[mateomy]

I know if you plug in the numbers straight out, like you have done it looks increasing. The thing is, I want to take the derivative and actually prove it. Which brings me back to my original problem: How to deal with the Factorial. As far as the Ratio test is concerned, the section we're covering is strictly Alternating series test, so we have to show it using that specific test. The ratio test is the next section we're covering so I don't think I can use it, though, I have an okay understanding of how to apply that one.

 

[Dick]

I know if you plug in the numbers straight out, like you have done it looks increasing. The thing is, I want to take the derivative and actually prove it. Which brings me back to my original problem: How to deal with the Factorial. As far as the Ratio test is concerned, the section we're covering is strictly Alternating series test, so we have to show it using that specific test. The ratio test is the next section we're covering so I don't think I can use it, though, I have an okay understanding of how to apply that one.

You can't really take the derivative of n!. Well, you can if you replace n! with a continuous gamma function, but I'm sure you haven't covered that. If the ratio test is in the next section then your only choice is to find an argument that n^n/n! is a increasing function of n without using derivatives. Start with writing it as the product (n/1)*(n/2)*(n/3)*...*(n/(n-1))*(n/n). When you go to the next term in the series you add 1 to the numerator of all of those factors and add an extra factor of (n+1)/(n+1). Can you argue that that will cause the product to increase?