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Math Help - Problem of sequence

  1. #1
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    Problem of sequence

    First of all, please my poor English (I'm currently a French student trying to master mathematical redaction in another language... And I do find it difficult.)
    So, I have an exercise I don't know how to handle...

    Let (Wn) be a sequence such as W_n= sum from (k=1 to n) of (1/(n+k)^1/2). Prove that for any integer n, (n/2)^1/2<= (Wn)
    I tried to prove it using induction:
    Let for any positive integer be Pn:" (n/2)^1/2<=(Wn)
    Basis: for n= 1, we have W_0= 1/(1+1)^1/2=1/(2)^1/2 , thus (P_0) is confirmed
    Then I assumed Pn, and tried to study W(n+1)
    Which I'm currently failing at...

    Any tips, any idea about how I could resolve this?

    Thanks for reading me! (And sorry for not using LaTEX, I'm currently trying to get accustomed to it... :/ )
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  2. #2
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    Re: Problem of sequence

    Each of n terms in W_n is at least \frac{1}{\sqrt{2n}} ...
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  3. #3
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    Re: Problem of sequence

    Hum, I assume you're right (yet I haven't worked out WHY each of the terms of Wn is greater than 1/(V(2n))... Since Pn only assume that Wn is greater than V(n/2). But I'll find the reason why anyway, thank you very much! )
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  4. #4
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    Re: Problem of sequence

    It's easier to give a direct, not an inductive, proof. \frac{1}{\sqrt{n+k}}\ge\frac{1}{\sqrt{2n}} because n+k\le2n.
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  5. #5
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    Re: Problem of sequence

    Oh yep, got it! Thank you, the rest followed from it quite easily!
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