Proving convergence for integral

In summary, the improper integrals evaluated from a to ∞ and 1 to ∞ for the functions exp(-at) and exp(-2at) converge to the value 1/aexp(a^2).
  • #1
zebo
14
0

Homework Statement


Prove that for every a ∈ ℝ+ the following improper integrals are convergent and measure its value.

∫a∞exp(-at)dt
Edited by mentor: ##\int_a^{\infty} e^{-at} dt##

∫1∞exp(-2at)dt
Edited by mentor: ##\int_1^{\infty} e^{-2at} dt##

The Attempt at a Solution



For the first integral i get -1/t+exp(t^2)+1/aexp(a^2) which for t going to infinity is convergent with the value 1/aexp(a^2)

For the second integral i get that it converges towards 1/2aexp(2a) for t going to infinity.

My issue is, that i am not quite sure how to explain, that this shows that for every a > 0 the integrals converges?
 
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  • #2
Dont know what went wrong when i posted, but the integrals are meant to be read as from a to ∞ and 1 to ∞ for the functions exp(-at) and exp(-2at)
 
  • #3
You need to show your work. For example, for the first one you need to show
$$\lim_{b\to \infty}\int_a^b e^{-at}~dt$$ is finite and give its value. And check your antiderivative. And did you mean to have ##a## both in the lower limit and the integrand?
 
  • #4
LCKurtz said:
You need to show your work. For example, for the first one you need to show
$$\lim_{b\to \infty}\int_a^b e^{-at}~dt$$ is finite and give its value. And check your antiderivative. And did you mean to have ##a## both in the lower limit and the integrand?

Yes a is both in the lower limit and in the integrand in the assignment, was confused by this at first as well. I assign t > a in the first one, and t > 1 in the second one, and then i let t go towards infinity to show that they are convergent with the values i found when t goes towards infinity. Is this enough though, to conclude that for every a > 0 they are convergent?

Also, the last part of the assignment is to solve the equation: I1 = I2 (I1 is the first integral and I2 is the second integral)
 
  • #5
Your first one is wrong and we have nothing to discuss until you show your work. It doesn't help us for you to just tell us "what you got". And even then, you need to put parentheses in the correct places for us to even know what you did get.
 
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  • #6
In your calculus book, this topic is called improper integrals. There are generally 2 types of improper integrals for your case. I can help you, but like others have said, you have to show some work.
 
  • #7
The first one, i assign t > a as the upper limit. When i solve it i get (-1/t)*(1/e^(t^2)) - (-1/a)*(1/e^(a^2)) = 1/(a*e^(a^2)) - 1/(t*e^(t^2)) . For t → ∞, the integral → 1/(a*e^(a^2))
 
  • #8
I don't know what you mean? I integrate it and get -1/a * exp(-at) = - 1/(a*exp(at)) which gives me the above answer when i use the upper and lower limit.
 
  • #9
zebo said:
I don't know what you mean? I integrate it and get -1/a * exp(-at) = - 1/(a*exp(at)) which gives me the above answer when i use the upper and lower limit.

But in post #1 you wrote "For the first integral i get -1/t+exp(t^2)+1/aexp(a^2)". I copied/pasted this directly from your message!
 
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  • #10
Ray Vickson said:
But in post #1 you wrote "For the first integral i get -1/t+exp(t^2)+1/aexp(a^2)". I copied/pasted this directly from you message!
Oh yeah, went alittle fast from my paper to here, sorry about that i ofcourse meant -1/(t*exp(t^2)) not 1/t+exp(t^2).

I am heading to bed, but i will work on it tomorrow. It is not so much the solving of the integrals, but more the understanding behind why this shows that for all positive values of a this shows that the integrals are convergent with above mentioned values. If the above shows the solution in which a takes part, then how do i explain / understand that this shows that for all positive values of a it is convergent?

On another note, i am asked to solve the equation 1/(a*exp(a^2)) = 1/(2a*exp(2a)) and i can't quite wrap my head around how to begin.

Thx in advance for any / all help and sorry if i am coming off as a guy who just wants you to solve my stuff, since this is not the case. If you have any link to someplace where i can understand this better it would be greatly appreciated.
 

What is the purpose of proving convergence for an integral?

The purpose of proving convergence for an integral is to ensure that the integral exists and has a finite value. This is important in many mathematical and scientific applications where the integral is used to calculate important quantities.

What is the definition of a convergent integral?

A convergent integral is one where the limit of the integral as the upper and lower bounds approach infinity is a finite value. In other words, the integral does not approach infinity or negative infinity as the bounds increase.

How is convergence for an integral typically proven?

Convergence for an integral is typically proven using mathematical techniques such as the comparison test, limit comparison test, or the integral test. These methods involve comparing the given integral to a known convergent or divergent integral, or evaluating the limit of the integral as the bounds approach infinity.

What are some common mistakes when trying to prove convergence for an integral?

Some common mistakes when trying to prove convergence for an integral include using the wrong test, not properly evaluating the limit of the integral, or incorrectly comparing the given integral to a known convergent or divergent integral. It is important to carefully follow the steps of the chosen test and ensure all calculations are correct.

What are some real-world applications of proving convergence for an integral?

Proving convergence for an integral is important in many fields such as physics, engineering, economics, and statistics. It is used to calculate important quantities such as area, volume, and probability. For example, in physics, the convergence of an integral is crucial in calculating the work done by a force or the total energy of a system.

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