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Math Help - Induction Problem

  1. #1
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    Induction Problem

    Prove the following statement:

    (For all natural numbers n)(Summation from k=1 to n (1/k>=ln(n+1))
    Some induction books use P(0) as the base case, in my book we use P(1), anybody kno how to approach?

    Also we may use the fact that ln(1+x)<= x whenever x belongs to real numbers and x >=0
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by ruprotein View Post
    Prove the following statement:

    (For all natural numbers n)(Summation from k=1 to n (1/k>=ln(n+1))
    Some induction books use P(0) as the base case, in my book we use P(1), anybody kno how to approach?

    Also we may use the fact that ln(1+x)<= x whenever x belongs to real numbers and x >=0
    Lets assume that the base case P(1) has been demonstrated.

    Suppose P(n) is true,m then:

    \sum_{k=1}^n 1/k \ge \ln(n+1)

    so:

    \sum_{k=1}^{n+1} 1/k = \sum_{k=1}^{n} 1/k + 1/(n+1) \ge \ln(n+1)+1/(n+1)

    Now:

    \frac{1}{n+1}\ge \int_n^{n+1}\frac{1}{x+1}dx=\ln(n+2)-\ln(n+1),

    so:


    \sum_{k=1}^{n+1} 1/k = \sum_{k=1}^{n} 1/k + 1/(n+1) \ge \ln(n+1)+\ln(n+2)-\ln(n+1)=\ln(n+2)

    which is what we need to complete the proof.

    RonL
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