Proving convergence of integral

In summary, the given double integral is convergent, as proven by using the direct comparison test with a convergent p-series. The steps taken in this proof were to first rewrite the integrand as a series, then to use the direct comparison test to show that it is smaller than a convergent p-series. It was also noted that the series needs to be convergent in order to pass through the integral, so the partial sums were taken to a finite value before taking the limit as it goes to infinity. Overall, the work presented was neat and accurate.
  • #1
Panphobia
435
13

Homework Statement


Prove the following double integral is convergent.

##\int_0^1 \int_0^1 \frac{1}{1-xy}\, dx \, dy##

The Attempt at a Solution


This was a bonus question on my final exam in calc 3 yesterday, I just want to show my steps and see if they were right.

So I realized that
##\frac{1}{1 - xy} = \sum_{n=1}^\infty x^ny^n##

So
##\int_0^1 \int_0^1 \sum_{n=1}^\infty x^ny^n\, dx \, dy = \sum_{n=1}^\infty\frac{1}{(n+1)^2}##
Then I used the direct comparison test to show that

##\frac{1}{(n+1)^2} \lt \frac{1}{n^2}##
So since it is smaller than a convergent p-series, it is also convergent.
 
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  • #2
Thanks for sharing this problem.
My only note would be that (I think) a series needs to be convergent in order to pass it through the integral. I would first take the partial sum to some large (finite) M, then you can pass it through and take the limit of the partial sums as M goes to infinity.

Aside from that, your work looks good. Nice and clean.
 

1. What is the definition of convergence of an integral?

The convergence of an integral refers to the idea that as the limits of integration approach infinity, the value of the integral approaches a finite number. In other words, the integral is said to converge if the area under the curve can be determined with a finite value.

2. How do you prove convergence of an integral?

To prove convergence of an integral, there are various methods such as the comparison test, limit comparison test, and the integral test. These methods involve evaluating the integral and determining if it meets certain criteria to show that it converges.

3. What is the importance of proving convergence of an integral?

Proving convergence of an integral is important because it allows us to determine the area under a curve and calculate important quantities such as volume, distance, and work. It also helps in solving more complex mathematical problems that involve integrals.

4. What happens if an integral does not converge?

If an integral does not converge, it is said to diverge. This means that as the limits of integration approach infinity, the value of the integral does not approach a finite number. In other words, the area under the curve cannot be determined with a finite value.

5. How do you determine if an integral is divergent?

To determine if an integral is divergent, one can use the comparison test or the limit comparison test. These methods involve comparing the given integral to a known integral that either converges or diverges. If the known integral diverges, then the given integral is also divergent.

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