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How would i evaluate an integral such as this

  1. Jun 20, 2009 #1
    1. The problem statement, all variables and given/known data
    ∫ (a^(n+1)-1)/(a-1) da

    2. Relevant equations
    i took this equation from the series of sigma with the bound of N k=0 and the argument as a^K (I'm sorry if this is confusing i don't know how to type it out and I'm not sure what all the correct terminology is).

    3. The attempt at a solution
    i really just don't know where to start with this one, i don't know what would be appropriate to use, by parts or substitution, and if it is by substitution what do i substitute?
  2. jcsd
  3. Jun 20, 2009 #2
    Are you saying that the original problem was

    [tex]\int \left( \sum_{k=0}^{n} {a^k} \right) da [/tex]


    In that case, you can use the fact that ∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx and apply this over and over again to your summation.

    The summation implies that n is a nonnegative integer, but if that is not the case, I don't think this function has an elementary integral.
  4. Jun 20, 2009 #3
    what is a non-elementary integral?
    and basically i was just trying to see what would happen if the summation was switched with an integral, because that sum is (when n approaches infinite) is 2. But that's only when the numbers increase at an integer amount, i wanted to see what it would be if it increased at every amount so i thought substituting the integral in would work. (is my logic flawed)

    EDIT:forgot to put in the fact that it only approaches two when a=1/2
    Last edited: Jun 20, 2009
  5. Jun 20, 2009 #4


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    [itex]a^{n+1}- 1= a^{n+1}- 1^{n+1}= (a-1)(a^n+ a^{n-1}+\cdot\cdot\cdot+ 1)[/itex]. Does that help?
  6. Jun 21, 2009 #5





    Is it enough?

  7. Jun 21, 2009 #6
    A non-elementary integral is one that cannot be expressed in elementary functions.

    An elementary function is one that can be written in terms of "simple" functions. A more thorough definition can be found at http://mathworld.wolfram.com/ElementaryFunction.html

    So you started with

    [tex] \lim_{n \to \infty} \sum_{k=0}^{n} {\left( \frac{1}{2} \right)^k} = 2 [/tex]

    right? I'm sorry, but I don't understand what you did afterwards.
  8. Jun 23, 2009 #7
    after that i replaced the sigma notation with an integral
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