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Math Help - sequences

  1. #1
    Senior Member sfspitfire23's Avatar
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    sequences

    I'm studying for a test and am looking at 2 questions which I'd like guidance with


    Suppose the sequence c_n converges to C. Define the sequence  d_n=\frac{c_n+c_{n+1}}{2} Does the sequence \{d_n\} converge or diverge?



    and the second Q

    Prove if \{a_n\} converges to a nonzero real number A, then there exists N such that |a_n|\geq \frac{1}{2}|A| for all n>N



    Thanks guys!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by sfspitfire23 View Post
    I'm studying for a test and am looking at 2 questions which I'd like guidance with


    Suppose the sequence c_n converges to C. Define the sequence  d_n=\frac{c_n+c_{n+1}}{2} Does the sequence \{d_n\} converge or diverge?
    If c_n converges to c then every subsequence ( c_{n+1} in particular) converges to c


    [quote



    and the second Q

    Prove if \{a_n\} converges to a nonzero real number A, then there exists N such that |a_n|\geq \frac{1}{2}|A| for all n>N



    Thanks guys![/QUOTE]

    Since A\ne 0\implies |A|>0\implies \frac{1}{2}|A|>0 we may take \varepsilon=\frac{1}{2}|A|.....so
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  3. #3
    Senior Member sfspitfire23's Avatar
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    Ah, so |a_n-A|<\frac{1}{2}|A| and the answer follows by adding A to -\frac{1}{2}|A|
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by sfspitfire23 View Post
    Ah, so |a_n-A|<\frac{1}{2}|A| and the answer follows by adding A to -\frac{1}{2}|A|
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