Find c likes the series is convergent

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The discussion focuses on determining the value of c ∈ IR for which the integral ∫₀^∞ (2x/(x²+1) - c/(2x+1)) dx converges. Participants suggest integrating the two components separately and analyzing their convergence behavior. Key insights include the necessity for the denominator to grow faster than the numerator for convergence, and the importance of finding a common denominator to simplify the expression. Ultimately, the discussion emphasizes the relationship between the growth rates of the numerator and denominator in integral convergence.

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Find c ∈ IR, like ∫∞ ( 2x - c ) dx is convergent
0 X^2 +1 2x+1

I need your help because I was trying to resolve the problem, but I couldn't, is difficult for me.
Please help me!
 
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Welcome to PF!

Hi marrie! Welcome to PF! :smile:
marrie said:
Find c ∈ IR, like ∫∞ ( 2x - c ) dx is convergent
0 X^2 +1 2x+1

hmm :rolleyes: … you mean \int_0^{\infty}\left(\frac{2x}{x^2+1} - \frac{c}{2x+1}\right)dx :wink:

Hint:

i] can you integrate each of them separately? does each converge or diverge?

ii] how much faster do you think the bottom needs to increase than the top for the integral to converge? :smile:
 
Ok, if I integrate each of them separately, I believe that the series diverge.
And I know that the bottom needs to increase than for the top for the integral converge.
thanks for your hints!
 
Hi marrie! :smile:
marrie said:
Ok, if I integrate each of them separately, I believe that the series diverge.
And I know that the bottom needs to increase than for the top for the integral converge.

ah … but how much more than the top? :wink:​

anyway, write the whole thing over a common denominator …

then what does c have to be to make the top increase slowly enough? :smile:
 

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