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Math Help - Summation

  1. #1
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    Summation

    I'm having trouble figuring this one out:

    <br />
lim_{n->infinity}\sum\limits_{i = 1}^{\color{red}n} {\frac{{f\left( {x_i } \right)}}{n}}<br />

    when f(x)=1/x between the interval [1,2]

    I keep on getting 1/infinity and that's zero, but the correct answer is appr. .693. thank you

    Note: its actually the summation of f(xsubi)*change in x but (2-1)/n is 1/n so i simplified it
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  2. #2
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    What is .693?. It's ln(2). See now?.
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  3. #3
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    no i don't unfortunately
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  4. #4
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    Quote Originally Posted by jarny View Post
    no i don't unfortunately
    The limit of your summation is  \ln{|2|} .

    Which just SO HAPPENS to be the result of the integral:

     \int_{1}^{2}{\frac{1}{x}\,dx}

    In other words:

     \displaystyle \lim_{n \to \infty} \sum_{i = 1}^{n} \frac{f(x_i)}{n} = \int_{1}^{2}{\frac{1}{x}\,dx}

    Such that  x_i \in \big[1,2\big]

    In general:

     \displaystyle \lim_{n \to \infty} \sum_{i = 1}^{n} f(x_i)\Delta_i = \int_{a}^{b}{f(x)\,dx}  for  x_i \in \big[a,b\big] .

    This is definition of a definite integral. Look it up. It's called the Riemann sum.
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  5. #5
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    I wasn't looking for the integral, that's easy. I was trying to do it via the formula i set out. I was just trying to do it via limits and i was unable to.
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