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?
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?
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.