(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

2. Relevant equations

Anything from calc 1 and 2

3. 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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# Homework Help: F(x) continuous, lim integral x to 2x of f(t) = 0 does integral of f(x)dx converge

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