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Math Help - Series/logarithm questions

  1. #1
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    Series/logarithm questions

    Hi, I was wondering how to solve these 2 questions:

    c) use the proof from part a) and d) [e^x=>x+1 and (1+1)(1+1/2)...(1+1/n)=n+1] to prove that e^(1+1/2+1/3+...+1/n)>n
    d) Find a value of n for which 1+1/2+1/3+...+1/n>100

    You don't have to provide the entire answer, but some hints or part of the work would be nice.

    Thanks in advance!
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Series/logarithm questions

    Having established that:

    (1+1)\left(1+\frac{1}{2} \right)\left(1+\frac{1}{2} \right)\cdots\left(1+\frac{1}{n} \right)=n+1

    and:

    e^x\ge x+1

    we may write:

    e^{1}\ge 1+1

    e^{\frac{1}{2}}\ge 1+\frac{1}{2}

    e^{\frac{1}{3}}\ge 1+\frac{1}{3}

    \vdots

    e^{\frac{1}{n}}\ge 1+\frac{1}{n}

    Since all values are positive, we may multiply down. What do you get?

    d) Having established that:

    e^{1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}}> n

    this implies:

    1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}>\ln(n  )

    Now, let:

    \ln(n)=100

    What does this say about n?
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  3. #3
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    Re: Series/logarithm questions

    Oh...thanks so much. I can't believe I didn't think that way for c) disappointed in myself lol
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