How can I efficiently test for convergence in integrals without wasting time?

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SUMMARY

This discussion focuses on efficiently testing for convergence in integrals, particularly distinguishing between elementary and non-elementary functions. A key strategy is to utilize comparison tests when faced with difficult integrals, as this can save time and effort. Participants emphasize that if integration attempts are unsuccessful after a few tries, it is prudent to consider the function as potentially non-elementary and seek simpler functions for comparison. This approach streamlines the process of determining convergence or divergence without unnecessary calculations.

PREREQUISITES
  • Understanding of integral calculus and convergence tests
  • Familiarity with elementary and non-elementary functions
  • Knowledge of comparison tests in calculus
  • Basic skills in mathematical problem-solving
NEXT STEPS
  • Research the Comparison Test for convergence in integrals
  • Study techniques for identifying elementary vs. non-elementary functions
  • Explore advanced convergence tests such as the Limit Comparison Test
  • Practice problems involving convergence of improper integrals
USEFUL FOR

Students studying calculus, particularly those focusing on integral convergence, as well as educators seeking effective teaching strategies for this topic.

MozAngeles
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Homework Statement


Hi, So i don't know if this is a stupid question but i'll ask anyways. So I'm on the chapter where we start testing integrals for convergence. The books starts out with elementary functions then they move towards non elementary functions. Testing for them is OK, my problem is that I cannot tell straight off the bat whether the integral is elementary, i waste a lot of time starting out trying to integrate. So is there any advice someone can give me that can help me identify my problem before i waste all the time?

Homework Equations





The Attempt at a Solution

 
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If you can't integrate a function after a few tries, start thinking it's maybe not elementary and try to find a comparison test with something that is. In fact, even if a function IS elementary, if you only have to show convergence or divergence it's often easier to prove that by comparison with functions that are easier to integrate than to do the original integral.
 
thanks. i realized after all these hours of studyin'
 

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