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Math Help - Proof of convergence

  1. #1
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    Proof of convergence

    Prove from the definition that the sequences of the form

    (An + B)/(Cn^2 + D)

    where A,B,C,D 2 R,C > 0,D 0
    are all convergent.

    Don't really know where to go with this...my teacher didn't explain how to prove convergence well, so I'm pretty stuck
    All I have is that I should prove by showing there exists a G st. for all epsilon > 0 there is an N > 0 st. for all n>N |(An+B)/(Cn^2+D) -G| < epsilon
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mistykz View Post
    Prove from the definition that the sequences of the form

    (An + B)/(Cn^2 + D)

    where A,B,C,D 2 R,C > 0,D 0
    are all convergent.

    Don't really know where to go with this...my teacher didn't explain how to prove convergence well, so I'm pretty stuck
    All I have is that I should prove by showing there exists a G st. for all epsilon > 0 there is an N > 0 st. for all n>N |(An+B)/(Cn^2+D) -G| < epsilon
    Since all the coefficients are all positive we see that \left|\frac{An+B}{Cn^2+D}\right|=\frac{An+B}{Cn^2+  D}\leqslant\frac{An}{Cn^2}=\frac{A}{C}\frac{1}{n}. Thus,for any \varepsilon>0 we can make \left|\frac{An+B}{Cn^2+D}\right|<\varepsilon by choosing n>\frac{1}{\frac{A}{C}\varepsilon}
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  3. #3
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    That makes sense...but I think I typo'd this:
    where A,B,C,D 2 R,C > 0,D 0
    should read
    where A,B,C,D are elements of R, C > 0, D>= 0

    so the all positives aren't necessarily given...only for the denominator
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