Mampac
New member
- Joined
- Nov 20, 2019
- Messages
- 48
Hello, could you correct my steps?
I'm dealing with
1) for convenience, I'm splitting the integral to two integrals from 0 to 1 and from 0 to +inf.
First of all, is there any way to simplify [MATH]arctan(sinx)[/MATH]? I searched all the internet but there seems to be no way of doing that.
2) I'm assuming it's convergent, because [MATH]arctan(sinx)[/MATH] is bounded and [MATH]x^2[/MATH]'s power is bigger than 1.
3) For Direct Comparison Test, I have to find a function bigger that's convergent. I make the numerator bigger by replacing [MATH]arctan(sinx)[/MATH] with [MATH]{\dfrac{π}{2}}[/MATH] (which I'll take out of the integral later) and make the denominator smaller by dropping [MATH]3cosx + 4[/MATH].
4) I prove that the integral from 1 to +inf of [MATH]{\dfrac{1}{x^2}}[/MATH]is convergent since x's power is greater than 1.
However, I do have many doubts about it! Is this enough? I look at the definition, but it seems like something's missing (((
I tried looking at the Limit Comparison Test, but I'm failing to solve the limit:
5) I see I can pull out [MATH]x^2[/MATH] in the denominator and get rid of the cosine and 4, but what do I do with the arctangent? if i just ignore it I get again "something divided by [MATH]x^2[/MATH]. Is this gonna be enough to prove the convergence?
Thank you in advance <3
I'm dealing with
1) for convenience, I'm splitting the integral to two integrals from 0 to 1 and from 0 to +inf.
First of all, is there any way to simplify [MATH]arctan(sinx)[/MATH]? I searched all the internet but there seems to be no way of doing that.
2) I'm assuming it's convergent, because [MATH]arctan(sinx)[/MATH] is bounded and [MATH]x^2[/MATH]'s power is bigger than 1.
3) For Direct Comparison Test, I have to find a function bigger that's convergent. I make the numerator bigger by replacing [MATH]arctan(sinx)[/MATH] with [MATH]{\dfrac{π}{2}}[/MATH] (which I'll take out of the integral later) and make the denominator smaller by dropping [MATH]3cosx + 4[/MATH].
4) I prove that the integral from 1 to +inf of [MATH]{\dfrac{1}{x^2}}[/MATH]is convergent since x's power is greater than 1.
However, I do have many doubts about it! Is this enough? I look at the definition, but it seems like something's missing (((
I tried looking at the Limit Comparison Test, but I'm failing to solve the limit:
5) I see I can pull out [MATH]x^2[/MATH] in the denominator and get rid of the cosine and 4, but what do I do with the arctangent? if i just ignore it I get again "something divided by [MATH]x^2[/MATH]. Is this gonna be enough to prove the convergence?
Thank you in advance <3