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Math Help - Proof Question sequences

  1. #1
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    Proof Question sequences

    Hello.

    I am having trouble proving this problem.

    Let (xn) be a sequence of positive numbers such that lim [(xn)^(1/n)] exists and equals a. a is greater than 0, but less than 1. Prove that lim (xn) = 0.

    The limits are to inf.

    Thank you.
    Timothy.
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  2. #2
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    If 1>r>0 then the sequence r^(1/n) converges to 1.
    So if glb(x_n)>0 that would contradict the given that a<1.
    Now use the fact the glb(x_n)=0 to prove the statement.
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  3. #3
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    Quote Originally Posted by THulchenko View Post
    Hello.

    I am having trouble proving this problem.

    Let (xn) be a sequence of positive numbers such that lim [(xn)^(1/n)] exists and equals a. a is greater than 0, but less than 1. Prove that lim (xn) = 0.

    The limits are to inf.

    Thank you.
    Timothy.
    If
    0<lim[(xn)^(1/n)]<1
    Then by the root test,
    The sequence of partial sums converges,
    SUM xn -----> Converges.
    That means the sequence,
    {xn}----> 0
    Because the sequence of the series always converges to zero whenver the infinite series exists.
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