Thread: 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?

No. The sum 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. , , , etc. but there is no general formula.

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?

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 .

Is it proven that does not exist a general formula, which are conditions?
Why does not exist a general formula...?

Originally Posted by Emilijo

Why does not exist a general formula...?

The sum 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 (note that the sum satisfies 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.