Comparing Integrals: Test for Convergence/Divergence

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SUMMARY

The integral from 1 to infinity of (2 + e^(-x))/x is confirmed to be divergent using the comparison test. The comparison is made with the integral of (2/x), which is known to diverge. Since (2 + e^(-x))/x is greater than (2/x) for all x ≥ 1, the original integral also diverges. This conclusion is definitive based on established mathematical principles.

PREREQUISITES
  • Understanding of integral calculus, specifically improper integrals.
  • Familiarity with the comparison test for convergence and divergence of integrals.
  • Knowledge of exponential functions and their behavior as x approaches infinity.
  • Ability to analyze limits and asymptotic behavior of functions.
NEXT STEPS
  • Study the comparison test in detail, focusing on its application to improper integrals.
  • Learn about other convergence tests, such as the ratio test and root test.
  • Explore the properties of exponential decay and its impact on integrals.
  • Practice solving various improper integrals to strengthen understanding of convergence and divergence.
USEFUL FOR

Students studying calculus, particularly those focusing on integral convergence, as well as educators teaching integral calculus concepts.

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Homework Statement



Use comparison test to see if the integral is convergent or divergent.

Homework Equations



integral [1,∞] (2+e^(-x))/x



The Attempt at a Solution



My books says that (1+e^(-x))/x is divergent and since my integral is bigger it is divergent also.
TRUE OR FALSE? Thanks for the help
 
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Jbreezy said:

Homework Statement



Use comparison test to see if the integral is convergent or divergent.

Homework Equations



integral [1,∞] (2+e^(-x))/x



The Attempt at a Solution



My books says that (1+e^(-x))/x is divergent and since my integral is bigger it is divergent also.
TRUE OR FALSE? Thanks for the help

Compare your integral with ##\int_1^\infty \frac 2 x~dx##.
 
Your integral also Diverges. So it diverges I like yours better though.
I have another one I will post in another forum. In a bit. Thanks.
 

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