Testing for Divergence using the Integral Test

In summary, the conversation discusses the Integral Test for testing convergence and divergence of series. The speaker had trouble with a particular series and tried various methods before using the Integral Test. They integrated using integration by parts and realized that the resulting integral had u and v terms whose limits went to infinity. The speaker asks if showing that uv approaches infinity is enough to prove divergence, or if the improper integral of vdu also needs to be solved.
  • #1
NotGauss
24
6
Hello all,
I was working on some homework regarding testing for convergence and divergence of series and I was having trouble with a particular series (doesn't really matter which one) and tried almost all the methods; then tried the Integral Test, my series met the conditions of the Integral Test and I integrated using integration by parts. Once I started solving the integral, I usually leave the uv terms for last since that is the simple part, I realized that I had u and v terms (using this form ∫udv = uv - ∫vdu) whos limit went to infinity.
My understanding of the Integral Test is that if the resulting integral is divergent (not a finite sum) then the summation of the series is divergent as well. So, is it enough to show that uv approaches infinity to show that my original integral (∫udv) is divergent? Or do I need to solve the improper integral of ∫vdu as well?

Thanks for your time and help!

Edit: If so, i just thought that this would be an easy way to show divergence, as you wouldn't have to fully calculate the integral but merely calculate the u and v terms.
 
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  • #2
You need to compute ##\int udv## as well, since if it is infinite it might cancel ##uv##.
 

FAQ: Testing for Divergence using the Integral Test

1. What is the Integral Test and how does it work?

The Integral Test is a method used to determine the convergence or divergence of an infinite series. It involves comparing the series to a corresponding improper integral and using the properties of integrals to determine the convergence or divergence of the series.

2. When should the Integral Test be used?

The Integral Test should be used when the series in question is positive, continuous, and decreasing. It is also useful when the series involves rational functions or exponential functions.

3. What is the process for using the Integral Test?

The process for using the Integral Test involves evaluating the corresponding improper integral and then comparing it to the original series. If the integral converges, then the series also converges. If the integral diverges, then the series also diverges.

4. Are there any limitations to the Integral Test?

Yes, there are some limitations to the Integral Test. It can only be used for positive series and cannot determine the exact value of the sum of a series. It also cannot be used for series with alternating signs.

5. How accurate is the Integral Test?

The Integral Test is a very accurate method for determining the convergence or divergence of a series. However, it may not work for all series and it is important to check for any exceptions or limitations before using it.

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