Does This Infinite Series Converge or Diverge?

In summary, the conversation discusses various problems and methods for proving convergence or divergence of series and finding the Maclaurin series of a given function. The first two problems involve using the integral test, comparison test, root test, or ratio test. The third problem gives a hint to explicitly use the Maclaurin series for cosine, which is 1-x^2/2+x^4/4-... To apply this to 3x^2cos(x^3), one must make the substitution x^3 in place of x. For the fourth problem, one must find the Maclaurin series for cos(x^3) by replacing all x's with x^3, which is 1-x^6/
  • #1
Frillth
80
0
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?
 
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  • #2
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?
 
  • #3
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).
 
  • #4
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)... 10x is 10*10*10... xx is x*x*x*x. The first thing you should note is that 10x/xx is 10/x*10/x*10/x

Then try spreading the x! terms over those terms (there are x terms in x!)
 
  • #5
[tex] \sum_{m=1}^{\infty} \frac{\sqrt{m+6}}{m^{2} + 2m}< \frac{\sqrt{m+6}}{m^{2}}\sim \frac{\sqrt{m}}{m^{2}} [/tex]For the first one use the ratio test: note that [tex] (x+1)! = (x+1)x! [/tex]
 
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  • #6
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...
 
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  • #7
The Maclaurin Series for [tex] \cos x [/tex] is [tex] \cos x = \sum_{n=0}^{\infty} (-1)^{n} \frac{x^{2n}}{2n!} [/tex]

What would it be for [tex] \cos x^{3} [/tex]?
 

Related to Does This Infinite Series Converge or Diverge?

1. What is an infinite series problem?

An infinite series problem is a type of mathematical problem that involves an infinite sum of numbers or terms. It can be represented as a sequence of terms that are added together, with each term becoming smaller and smaller as the series continues.

2. How do you determine if an infinite series converges or diverges?

To determine if an infinite series converges or diverges, you can use different tests such as the comparison test, ratio test, or integral test. These tests evaluate the behavior of the series and can help determine if it approaches a finite value (converges) or if it goes to infinity (diverges).

3. What is the difference between a convergent and divergent infinite series?

A convergent infinite series is one in which the terms approach a finite value as the series continues. This means that the sum of the series will also approach a finite value. In contrast, a divergent infinite series is one in which the terms do not approach a finite value and the sum of the series goes to infinity.

4. How can infinite series problems be applied in real-life situations?

Infinite series problems have many applications in various fields such as physics, engineering, and finance. For example, they can be used to model the behavior of electrical circuits, the growth of bacteria, or the value of an investment over time.

5. What are some common techniques for solving infinite series problems?

Some common techniques for solving infinite series problems include finding patterns in the terms, using formulas such as the geometric series formula, and applying known convergence tests. Computer software and calculators can also be useful tools for solving more complex series problems.

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