Does Cauchy Test Fail If $\lim_{x\to\infty}\int_x^{2x}f(t)dt = 0$?

In summary, the conversation discusses the convergence of an integral when given certain conditions. The participants explore the possibility of finding a function that does not converge, but also satisfies the condition of having a limit of the integral from x to 2x being equal to 0. After some discussion, they come to the conclusion that such a function could be a harmonic series.
  • #1
no_alone
32
0

Homework Statement


True Or False
if f(x) continuous in [tex][a,\infty][/tex] and [tex]\lim_{x\to\infty}\int_x^{2x}f(t)dt = 0 [/tex]
Then [tex]\int_a^\infty f(x)dx [/tex] converge


Homework Equations


Anything from calc 1 and 2


The Attempt at a Solution


Actually I'm really stuck..
My main motive is to try and show that if [tex]\lim_{x\to\infty}\int_x^{2x}f(t)dt = 0 [/tex] there is no way that Cauchy test can fail,
I assume Cauchy test fail and that [tex]\lim_{x\to\infty}\int_x^{2x}f(t)dt = 0 [/tex] and try to show that there is no option for this to happen.
First I tried on f(x) that can be positive and negative .. didn't made it, Then I tried on f(x) that is only positive also didn't made it..
Maybe its false, But I didn't found any function that does not converge and also [tex]\lim_{x\to\infty}\int_x^{2x}f(t)dt = 0 [/tex]
Thank you.
 
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  • #2


Well, let's try and make such a function. Suppose we make the integral over [a,2a]~1/2, the integral over [2a,4a]~1/3, the integral over [4a,8a]~1/4. Then integral over all t should diverge like a harmonic series, right? Now, what must the values of such a function look like? Over [a,2a] it should be about 1/(2a). Over [2a,4a] it should be about 1/(3*(2a)). Over [4a,8a] like about 1/(4*(4a)). Etc, etc. That sort of looks like 1/(log(t)*t) to me.
 
Last edited:
  • #3


Thank you Dick
I didn't thought about series all I had in mind were function..
I tried and tried , even when I tried to build a function like you did by [a,2a],[4a,8a], I did it by building it by other function, I never imagine building a function with a series
Thank you.
 

Related to Does Cauchy Test Fail If $\lim_{x\to\infty}\int_x^{2x}f(t)dt = 0$?

Question 1: What is the Cauchy Test and how is it used?

The Cauchy Test is a method used to test the convergence of an infinite series. It states that if the limit of the ratio of consecutive terms is equal to zero, then the series converges.

Question 2: How does the Cauchy Test relate to the integral test?

The Cauchy Test is closely related to the integral test, as both methods are used to determine the convergence of an infinite series. The integral test uses the comparison of the infinite series with an integral, while the Cauchy Test looks at the ratio of consecutive terms.

Question 3: What is the significance of $\lim_{x\to\infty}\int_x^{2x}f(t)dt = 0$ in the Cauchy Test?

This equation represents the condition for the Cauchy Test to fail. If the integral of the function f(t) from x to 2x approaches zero as x approaches infinity, the Cauchy Test cannot be used to determine the convergence of the series.

Question 4: Can the Cauchy Test be used for all infinite series?

No, the Cauchy Test is not applicable for all infinite series. It can only be used for series with positive terms, and it may fail for series that have alternating terms or negative terms.

Question 5: Are there any other tests that can be used to determine the convergence of an infinite series if the Cauchy Test fails?

Yes, there are several other tests that can be used if the Cauchy Test fails. Some examples include the ratio test, the root test, and the comparison test.

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