Comparing Integrals: Test for Convergence/Divergence

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In summary, the conversation discusses using the comparison test to determine if the integral [1,∞] (2+e^(-x))/x is convergent or divergent. The attempt at a solution involves comparing the integral with another one and concluding that it also diverges.
  • #1
Jbreezy
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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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  • #2
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##.
 
  • #3
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.
 

What is the purpose of comparing integrals to test for convergence/divergence?

The purpose of comparing integrals is to determine whether a given series of functions converges or diverges. This is important in mathematical analysis and can help us understand the behavior of a function as the input values approach certain limits.

What is the test for convergence/divergence of integrals?

The test for convergence/divergence of integrals is a method for determining the behavior of a function as its input values approach certain limits. It involves comparing the integral of a given function to the integral of a known function, such as a geometric series, to determine if the original function converges or diverges.

What is the relationship between the convergence/divergence of integrals and series?

The convergence or divergence of an integral is closely related to the convergence or divergence of the corresponding series. In fact, the integral test is often used to prove the convergence or divergence of a series.

What are some common functions used for comparison in the test for convergence/divergence of integrals?

Some common functions used for comparison in the test for convergence/divergence of integrals include geometric series, p-series, and harmonic series. These functions have well-known convergence or divergence behaviors, making them useful for comparison.

What are some potential limitations of using the test for convergence/divergence of integrals?

One potential limitation of using the test for convergence/divergence of integrals is that it may not always provide a definitive answer. In some cases, the test may be inconclusive or require additional methods to determine the convergence or divergence of a function. Additionally, the test may not be applicable to all types of functions, such as those with discontinuities or oscillating behavior.

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