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Math Help - analysis problem

  1. #1
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    Jun 2009
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    9

    analysis problem

    For a positive integer
    n , let
    an=1+(1/2)+(1/3)+..........+(1/(2^n -1))
    then
    a) a200<=100
    b) a200 >100

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  2. #2
    Member
    Joined
    Jun 2009
    Posts
    113
    Here it is underlying the original proof of the divergence of the harmonic series.

    Define
    s_1=1>1/2
    s_2=1/2
    s_3=1/3+1/4>2(1/4)=1/2
    s_4 =1/5+1/6+1/7+1/8>4(1/8)=1/2
    s_5=1/9+\cdots+1/16>8(1/16)=1/2

     <br />
s_n=(1/(2^{n-2}+1)+\cdots+1/2^{n-1}>(2^{n-1}-2^{n-2})1/2^{n-1}=1/2<br />

     <br />
a_n=s_1+\cdots+s_{n}>n(1/2)<br />

    Now it follows that the harmonic series diverges, and also that your correct answer is b)!
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