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Math Help - Proving that limit of sequence does not exists

  1. #1
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    Proving that limit of sequence does not exists

    Hello,

    I have some probelm regarding my homework.

    I proved that the sequence (-5)^{n} doesn't have limit and now I need to prove that the following sequence doesn't have limit too:
    \lim_{n \to \infty} \frac{(-5)^{n+1} + 5^n}{(-2)^n + 3}
    I tried to disprove it by the definition but it's doesn't go well so far, also I think it have to do something with the part that the sequence (-5)^{n} doesn't have limit.

    What is the best way to prove that this sequence doesn't have limit?
    Thanks for your help.
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  2. #2
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    Re: Proving that limit of sequence does not exists

    Quote Originally Posted by kaze1 View Post
    Hello,

    I have some probelm regarding my homework.

    I proved that the sequence (-5)^{n} doesn't have limit and now I need to prove that the following sequence doesn't have limit too:
    \lim_{n \to \infty} \frac{(-5)^{n+1} + 5^n}{(-2)^n + 3}
    I tried to disprove it by the definition but it's doesn't go well so far, also I think it have to do something with the part that the sequence (-5)^{n} doesn't have limit.

    \frac{(-5)^{n+1} + 5^n}{(-2)^n + 3}=\frac{(-1)^{n+1}(5)^{n+1} + 5^n}{(-1)^n(2)^n + 3}=\frac{(-1)^{n+1}(5) + 1}{(-1)^n\left(\tfrac{2}{5}\right)^n + 3\left(\frac{1}{5}\right)^n}

    Now what happens for n even or odd?
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  3. #3
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    Re: Proving that limit of sequence does not exists

    Jello, kaze1!

    Another approach . . .


    \lim_{n \to \infty} \frac{(\text{-}5)^{n+1} + 5^n}{(\text{-}2)^n + 3}

    We have: . \frac{\text{-}5(\text{-}5)^n + 5^n}{(\text{-}2)^n + 3} \;=\;\frac{\text{-}5(\text{-}1)^n(5^n) + 5^n}{(\text{-}1)^n(2^n) + 3} \;=\;\frac{5^n\big[1 - 5(\text{-}1)^n\big]}{3 + (\text{-}1)^n(2^n)}

    Divide numerator and denominator by 2^n:\;\;\frac{\left(\frac{5}{2}\right)^n\big[1 - 5(\text{-}1)^n\big]}{\frac{3}{2^n} + (\text{-}1)^n}

    Then: . \lim_{n\to\infty}\frac{\left(\frac{5}{2}\right)^n \big[1 - 5(\text{-}1)^n\big]}{\frac{3}{2^n} + (\text{-}1)^n} \;=\;\frac{\infty\big[1-5(\text{-}1)^n\big]}{0 + (\text{-}1)^n}

    . . Even n:\;\;\frac{\infty(1-5)}{1} \:=\:-\infty

    . . Odd n: \;\;\frac{\infty(1+5)}{-1} \:=\:-\infty


    The limit diverges to -\infty.
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