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



    and the second Q

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



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


    [quote



    and the second Q

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



    Thanks guys![/QUOTE]

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