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Math Help - Convergence by comparison

  1. #1
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    Convergence by comparison

    Can someone help me prove that the following integral converges or diverges by comparison?

    S 1/sqrt(1+x^2) dx from 0 to infinity


    Thank you
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  2. #2
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    \int_{0}^{\infty }{\frac{dx}{1+x^{2}}}=\int_{0}^{1}{\frac{dx}{1+x^{  2}}}+\int_{1}^{\infty }{\frac{dx}{1+x^{2}}}, since the first piece is continuous in [0,1] thus is integrable there (the integral exists, which means that it converges), now, for the second piece, we have that for x\ge1 it's \frac{1}{1+x^{2}}<\frac{1}{x^{2}}, since \int_1^{\infty}\frac{dx}{x^2}<\infty, then so does \int_{1}^{\infty }{\frac{dx}{1+x^{2}}}, whereat the whole integral converges.
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