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Math Help - Comparison Theorem

  1. #1
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    Comparison Theorem

    Not sure if I am using this theorem correctly...

    To evaluate x^2/(1+x^6) from negative infinity to positive infinity, I used x^2/x^6 as the comparison. After integrating, I took the limit from t to 0 as t -> - inf and the limit from 0 to s as s -> +inf of -1/(3x^3).

    So, if I have done this much correctly, would the limit be convergent with them both equal to 0? Or divergent with them both equal to infinity?
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  2. #2
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    Quote Originally Posted by veronicak5678 View Post
    Not sure if I am using this theorem correctly...

    To evaluate x^2/(1+x^6) from negative infinity to positive infinity, I used x^2/x^6 as the comparison. After integrating, I took the limit from t to 0 as t -> - inf and the limit from 0 to s as s -> +inf of -1/(3x^3).

    So, if I have done this much correctly, would the limit be convergent with them both equal to 0? Or divergent with them both equal to infinity?
    Let t=x^3.
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  3. #3
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    I'm sorry, I don't know what you mean. Do you mean the initial comparison should be to x^3?
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  4. #4
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    Quote Originally Posted by veronicak5678 View Post
    I'm sorry, I don't know what you mean. Do you mean the initial comparison should be to x^3?
    \int_{-\infty}^{\infty} \frac{x^2}{x^6+1} dx = \frac{1}{3}\int_{-\infty}^{\infty} \frac{(x^3)'}{(x^3)^2+1} dx = \frac{1}{3}\int_{-\infty}^{\infty} \frac{dx}{x^2+1} = \frac{\pi}{3}
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