Does This Infinite Series Converge or Diverge?

[Frillth]

I need help with the following problems:

1. Prove whether:
sum from x=1 to infinity of x!*10^x/x^x
converges or diverges
2. Prove whether:
sum from x=3 to infinity of sqrt(m+4)/(m^2-2m)
converges or diverges
3. Calculate the Maclaurin series of f(x)=3x^2*cos(x^3) Hint: Explicity use the Maclaurin series for cosine.
4. Using the series from 3, verify that the integral of 3x^2*cos(x^3)dx = sin(x^3) + C

For 1 and 2, I believe we're supposed to use the integral test, comparison test, root test, or ratio test. For 3 and 4, I'm not quite sure even how to start. We had about 15 problems of homework, but these are the only ones that are giving me trouble. Can anybody help me out?

 

[Hurkyl]

For 1 and 2, I believe we're supposed to use the integral test, comparison test, root test, or ratio test.

Well, have you tried any (or preferably, all) of them? If so, where are you having trouble?

For 3 and 4, I'm not quite sure even how to start.

Well, it gives you a big hint for #3. What is the MacLaurin series for cosine? Have you given any thought how you might use that for this problem?

 

[Frillth]

I've tried all of the methods for the first problem, but I still can't seem to figure it out. I'm not sure how to do limits with factorials or things like x^x. And for the second problem, I just can't seem to get a result with any test.

For 3/4, I know that the MacLaurin series for cosine is 1-X^2/2 +X^4/4..., but I don't know how to apply this to 3x^2cos(x^3).

 

[Office_Shredder]

What you posted is the series for cos(x). Try making that cos(x³) (this is as trivial as it can possibly be). Then multiply that by 3x²

For the second, start writing out terms. x! is x*(x-1)*(x-2)... 10^x is 10*10*10... x^x is x*x*x*x. The first thing you should note is that 10^x/x<sup>x is 10/x*10/x*10/x

Then try spreading the x! terms over those terms (there are x terms in x!)</sup>

 

[courtrigrad]

$$\sum_{m=1}^{\infty} \frac{\sqrt{m+6}}{m^{2} + 2m}< \frac{\sqrt{m+6}}{m^{2}}\sim \frac{\sqrt{m}}{m^{2}}$$For the first one use the ratio test: note that $$(x+1)! = (x+1)x!$$

 

[Frillth]

I now have the answers for 1 and 2, but I'm still having trouble with 3 and 4.

I'm not sure how to make cos(x) into cos(x^3). Is it like this:
1-x^8/8+x^64/64-x^216/216...?

I've never done any problems where you find one MacLaurin series based on another, so I'm kind of lost when you say it's trivial.

Edit: I think I got it for cos(x^3). For that, you would just turn all of the x's into x^3's, right? So it would be:

1-x^6/2+x^12/4-x^18/6...

 

[courtrigrad]

The Maclaurin Series for $$\cos x$$ is $$\cos x = \sum_{n=0}^{\infty} (-1)^{n} \frac{x^{2n}}{2n!}$$

What would it be for $$\cos x^{3}$$?