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Thread: Subsequences

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    Subsequences

    How many convergent subsequences, and how many limits of these subsequences are there in the following sequence:


    Justify your answer.
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  2. #2
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    Quote Originally Posted by matty888 View Post
    How many convergent subsequences, and how many limits of these subsequences are there in the following sequence:


    Justify your answer.
    An infinite amount...

    1/2 + 1/3 + 1/4 + 1/5 ....

    1/3 + 1/4 + 1/5 + 1/6 .....
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    Thanks(???)

    Thank you, could I justify this by writing in the form 1/n+1 and getting the sum to infinity??(Or something like that)
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    Quote Originally Posted by matty888 View Post
    Thank you, could I justify this by writing in the form 1/n+1 and getting the sum to infinity??(Or something like that)
    Yes, with different starting points.
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  5. #5
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    Quote Originally Posted by matty888 View Post
    How many convergent subsequences, and how many limits of these subsequences are there in the following sequence:


    Justify your answer.
    Every number in the interval [0,1] is the limit of a subsequence.

    Reason: Let \alpha\in[0,1]. For the n-th term of the subsequence, choose the term with denominator n and numerator k such that k/n comes closest to α. Then |\textstyle\frac kn-\alpha|<1/n. Hence the subsequence converges to α.
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