Use the integral test to determine if this series converges or diverges

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SUMMARY

The integral test was applied to determine the convergence of the series defined by the sum from n=1 to infinity of n/(1+(n^2)). The discussion highlighted the necessity of evaluating the improper integral corresponding to the series, which is essential for applying the integral test. Initial attempts using Symbolab were unsuccessful, indicating the need for a different approach. Ultimately, the solution involved using U-substitution to successfully evaluate the integral, confirming the series' behavior.

PREREQUISITES
  • Understanding of the integral test for series convergence
  • Familiarity with improper integrals
  • Knowledge of U-substitution in integral calculus
  • Basic concepts of series and sequences
NEXT STEPS
  • Study the integral test for series convergence in detail
  • Practice evaluating improper integrals using U-substitution
  • Explore other convergence tests such as the ratio test and comparison test
  • Learn how to use online integral calculators effectively
USEFUL FOR

Students in calculus courses, educators teaching series convergence, and anyone seeking to deepen their understanding of integral tests in mathematical analysis.

Randall
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Homework Statement


Use the integral test to determine if this series converges or diverges: sum from n=1 to infinity of n/(1+(n^2))

Homework Equations


Integral test: a series and it's improper integral both either converge or both diverge

The Attempt at a Solution


see attached - I need help finding the integral. I tried using an online integral calculator, symbolab but it says there is no integral. I'm guessing there is some way to split this up into pieces that I'm not seeing. Please help.
 

Attachments

  • integral_test.jpg
    integral_test.jpg
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Your proof that it is positive and decreasing isn't sufficient, but the integral is fairly trivial. U-substitution comes to mind.
 
Thank you! Yes I used substitution and got it to work :)
 

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