Solving 2 Tricky Math Problems: Bounded Area & Convergence of Sequence

In summary: I tried u= x^2+2 and got0\leq a_n \leq \frac{2}{n^3}. Think squeeze theorem.Thanx a lot. I tried ur hints and got some results. Can u guys check if they correct?I used the squeeze theorem to find the limit. The limit is 0, so the area is finite.
  • #1
yeuVi
12
0
There are two problems I got stuck...

1. Is the area in the first quadrant bounded between the x-axis and the curve

[tex]\ y= \frac{x} {2*(x^2+2)^{7/8}} [/tex]

finite? This one, I used the Area formula... but then I cannot integrate it... and then how to determine if it's finite or not?

2. Does the sequence

[tex]\ an= \frac{cos(n)+1}{n^3}[/tex]

converge?

How do you do this one? I need some hints to find the limit...

Thank a lot guys ^_^
 
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  • #2
1) Try using the comparison test with a simpler function that you can integrate (or know about it's convergence).

2)Hint: -1<=cos(n)<=1
 
  • #3
You can integrate this

[tex] \frac{1}{2}\int \frac{x}{\left(x^{2}+2\right)^{\frac{7}{8}}} \ dx [/tex]

Make the substitution

[tex] x^{2}+2=u [/tex]

Daniel.
 
  • #4
But the first one, after I get the integrate, what should I do next? I mean, it's is finite if the integrate limit is converge?

The secon one, I do not really understand the hint... can u explain a little more detail? I guess you want me to apply the comparion limit test for a sequence? I though that test is only for series...

Thanx every1 for helping me... ^_^
 
  • #5
yeuVi said:
But the first one, after I get the integrate, what should I do next? I mean, it's is finite if the integrate limit is converge?

Yes, if the integral converges you'd say the area was finite. If the integral diverges then you'd say the area was "infinite". (note since the integeand is positive there can only be one kind of divergence, growing without bound).

yeuVi said:
The secon one, I do not really understand the hint... can u explain a little more detail? I guess you want me to apply the comparion limit test for a sequence? I though that test is only for series...

[tex]0\leq a_n \leq \frac{2}{n^3}[/tex]. Think squeeze theorem.
 
  • #6
Thanx a lot. I tried ur hints and got some results. Can u guys check if they correct?

The 1st one, I got the area is finite...

The 2nd one, the sequence is convergence, since

[tex]\ \lim{0} = \lim{ \frac{2}{n^3}} =0 [/tex]

so [tex]\ \lim{a_n}=0[/tex]

therefore the sequence is converge to 0...
 
  • #7
yeuVi said:
The 1st one, I got the area is finite...

How about posting the details of your work?

yeuVi said:
The 2nd one, ... therefore the sequence is converge to 0...

Looks good.
 
  • #8
The 1st one, I integrated it by substitution u= x^2+2

so the integration is

1/4 * 8* (u^(1/8))

It's converge to 0, so the area 's finite...
 
  • #9
The function is always positive when x>0, so how could the area be 0?

HINT: What bounds did you integrate over? Remember, this is an area, so you should be taking a definite integral!
 
  • #10
Oh, it's not the area is 0, but the limit of the integral is 0...
 
  • #11
what limit of the integral?
 
  • #12
shmoe said:
Yes, if the integral converges you'd say the area was finite. If the integral diverges then you'd say the area was "infinite". (note since the integeand is positive there can only be one kind of divergence, growing without bound).

Oh, I was following his hint...
 
  • #13
The hint is fine, but I don't see how you found the integral to be convergent! Was the integral that you did

[tex]\int_0^\infty \frac{x}{2(x^2+2)^{\frac{7}{8}}} \ dx[/tex]

? Do you see why this is the correct integral, if not?
 

Related to Solving 2 Tricky Math Problems: Bounded Area & Convergence of Sequence

1. What is a bounded area in mathematics?

A bounded area in mathematics refers to a region or space that is limited or confined within a specific boundary. This means that the values or variables within the area do not exceed a certain range or limit.

2. How do you determine if a given area is bounded or not?

To determine if a given area is bounded, you can use various methods such as graphing the area and checking if it has a finite size, checking if the area has a maximum or minimum value, or using mathematical equations to set limits on the variables in the area.

3. What is the importance of understanding bounded areas in math?

Understanding bounded areas is crucial in many mathematical applications, such as optimization problems, integration, and differential equations. It allows us to set limits and constraints on variables, making it easier to solve complex problems and provide more accurate solutions.

4. What is the convergence of a sequence in mathematics?

In mathematics, the convergence of a sequence refers to the behavior of a sequence of values as it approaches a limit. It is the process of a sequence approaching a specific value or becoming closer to it as more terms in the sequence are added.

5. How do you determine if a sequence is convergent or divergent?

To determine if a sequence is convergent or divergent, you can use various methods, such as the limit comparison test, the ratio test, or the root test. These tests compare the given sequence to a known convergent or divergent sequence to determine its behavior.

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