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Thread: Power of sets

  1. #1
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    Power of sets

    HI,
    What is the power of the set of all sebsets of N on which the series ∑1/n converges to a limit smaller than 1.
    Thank's in advance.
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  2. #2
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    Re: Power of sets

    Hey hedi.

    What have you tried?
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  3. #3
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    Re: Power of sets

    Quote Originally Posted by hedi View Post
    HI,
    What is the power of the set of all subsets of N on which the series ∑1/n converges to a limit smaller than 1.
    What exactly is meant by the expression "subsets of N on which the series ∑1/n converges".
    Without knowing the indices of summation, it is total meaningless expression.
    Therefore, you need to be as complete as possible in your reply.
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    Re: Power of sets

    what is the power of the set of all possible summation indices.I need a clue or direction for a solution.
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    Re: Power of sets

    Quote Originally Posted by hedi View Post
    what is the power of the set of all possible summation indices.I need a clue or direction for a solution.
    Sorry but you reply means nothing to me.
    Please, give us several examples you are considering.
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  6. #6
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    Re: Power of sets

    For example,the set N in P(N) is not a member in our set (i.e the set that we want to find its cardinality),because the harmonic series diverges.The set of prime natural numbers may belongs because the series based on the prime indeces converges to a limit ,but i don't know if the limit is smaller than 1.

    see in google: "harmonic series and its parts".
    Last edited by hedi; Dec 24th 2016 at 05:11 PM.
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  7. #7
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    Re: Power of sets

    I think i solved it ,and the cardinality is the continuum ,that is,the cardinality of p(N).
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    Re: Power of sets

    Quote Originally Posted by hedi View Post
    For example,the set N in P(N) is not a member in our set (i.e the set that we want to find its cardinality),because the harmonic series diverges.The set of prime natural numbers may belongs because the series based on the prime indeces converges to a limit ,but i don't know if the limit is smaller than 1.
    That still is nonsense. Consider the subset of primes $K=\{2,3,5,7,11,13\}$
    $\sum\limits_{x \in K} {{x^{ - 1}}} = {\text{1}}{\text{.40285}}$ see HERE
    That finite sum is more than one so it won't do.
    Any finite subset of at least two terms one of which is one will not do. Why?

    Will the set $T=\{3^{-n}: n\in\mathbb{Z}^+\}$ do? Why or why not?
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