Convergence of Integral: How to Prove for 0<k<1?

In summary, the given problem involves showing that the integral of e^{kx}/(1+e^{x}) from negative infinity to positive infinity converges for 0<k<1. The attempt at a solution involves finding a dominating integral for both the intervals and proving their finiteness.
  • #1
rioo
6
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Homework Statement


Show that [itex]\int^{\infty}_{-\infty} \frac{e^{kx}}{1+e^{x}}dx[/itex] converges if [itex]0<k<1[/itex]


Homework Equations


None


The Attempt at a Solution


Well if I can show that the integral is dominated by another that converges then I'm done, but I haven't been able to come up with one. I've tried manipulating the integrand (moving the [itex]e^{kx}[/itex] to the bottom and checking limits. The integrand does go to zero at [itex]-\infty \mathrm{and\ } \infty[/itex], but that doesn't guarantee convergence...
 
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  • #2
rioo said:

Homework Statement


Show that [itex]\int^{\infty}_{-\infty} \frac{e^{kx}}{1+e^{x}}dx[/itex] converges if [itex]0<k<1[/itex]


Homework Equations


None


The Attempt at a Solution


Well if I can show that the integral is dominated by another that converges then I'm done, but I haven't been able to come up with one. I've tried manipulating the integrand (moving the [itex]e^{kx}[/itex] to the bottom and checking limits. The integrand does go to zero at [itex]-\infty \mathrm{and\ } \infty[/itex], but that doesn't guarantee convergence...

Look at the two cases ##\int_0^\infty## and ##\int_{-\infty}^0## separately and use different overestimates on the different intervals. If you can show they are both finite you are done.
 

1. What is meant by "show convergence of integral"?

"Show convergence of integral" refers to the process of determining whether an integral, or a mathematical expression used to calculate the area under a curve, has a finite value or not. In other words, it involves proving that the integral exists and is not infinite.

2. Why is it important to show convergence of integral?

Showing convergence of integral is important because it helps us determine the validity and accuracy of the integral. If an integral does not converge, it means that the area under the curve cannot be accurately calculated, and the integral cannot be used to solve mathematical problems.

3. What are the different methods used to show convergence of integral?

There are several methods used to show convergence of integral, including the comparison test, the limit comparison test, the integral test, and the ratio test. These methods involve comparing the integral to a known convergent or divergent series, or taking the limit of the integral to determine its behavior.

4. What are some common mistakes to avoid when showing convergence of integral?

Some common mistakes to avoid when showing convergence of integral include not checking the conditions of the convergence tests, using incorrect limits or bounds, and not properly evaluating the integral. It is important to carefully follow the steps of the chosen method and double-check all calculations to ensure accuracy.

5. Can an integral both converge and diverge?

No, an integral cannot both converge and diverge. It can only have one of these two possible outcomes. If the integral converges, it means that the area under the curve is finite and can be accurately calculated. If it diverges, it means that the area under the curve is infinite and cannot be accurately calculated.

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