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Math Help - Sum solving

  1. #1
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    Sum solving

    Is it possible to find F(n), summation of the function f(k), where k=1,2...,n ( Is there some formula, general formula, for any f(k), of course if k is in domain of function)

    sum_(k=1, to n) f(k) = F(n)

    If it is not possible, why?
    Last edited by Emilijo; July 16th 2012 at 02:53 PM.
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  2. #2
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    Re: Sum solving

    No. The sum f(1) + f(2) + \dots + f(n) is sometimes known as a Riemann sum. There is no general formula for a Riemann sum, other than approximating that sum with the integral of the function f(n) (assuming f is integrable).

    There are definitely formulas for specific functions f, e.g. f(n) = n, f(n) = n^2, f(n) = 3n+1, f(n) = \frac{1}{n(n+1)} etc. but there is no general formula.
    Last edited by richard1234; July 16th 2012 at 03:05 PM.
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  3. #3
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    Re: Sum solving

    But then how to solve this: sum_(k=1, to n) sin(1/k)
    Is it possible, I put it into wolfram alpha, but I didn t get the solution.
    Why is not possible to get it? Do exist conditions when the sum is possible to calculate, or is it just technical problem?
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  4. #4
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    Re: Sum solving

    Try this: \sum_{k=1}^{n} \sin(1/k). Or if you know how to type in Mathematica, you can do that as well.

    Obviously, you're not going to get a general solution. But note that \int_{1}^{n} \sin x \, dx = -\cos n + \cos 1.
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  5. #5
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    Re: Sum solving

    Is it proven that does not exist a general formula, which are conditions?
    Why does not exist a general formula...?
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  6. #6
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    Re: Sum solving

    Quote Originally Posted by Emilijo View Post
    Why does not exist a general formula...?
    The sum \sum_{k=1}^n\sin(1/k) is, of course, a well-defined function of n. However, in general there is no reason why some function has to be represented as a finite composition of a few basic functions and operations: addition, multiplication, sine, square root, etc. For example, the function that, given five coefficients of a quintic equation with the leading coefficient 1, returns the least real root of this equation, is also well-defined for all arguments. However, it is proven that this function cannot be expressed as a finite composition of the four arithmetic operations and roots of any degree.

    I would also like to know sufficient, or, better, necessary and sufficient conditions for when a recurrence relation x_{n+1}=g(x_n) (note that the sum x_n=\sum_{k=1}^nf(k) satisfies x_{n+1}=x_n+f(n+1)) has a closed-form solution. Well, there are sufficient conditions, such as when a relation is a linear homogeneous recurrence relation with constant coefficients, but they are not very general.
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