Can You Calculate the Derivative of a Function that is a Sum of n Terms?

[jimmy1]

Is it possible to calculate the derivative of a function of the form $$\sum_{i=1}^n f(i)$$ with respect to n.

That is, I have an function that is the sum of n terms and there is no closed form expression for it, and I need to get the derivative of the function w.r.t to n. I've tried to do it in Mathematica but just get error messages all the time, so is there any way to get the derivative w.r.t n of such a function?

 

[lurflurf]

Is it possible to calculate the derivative of a function of the form $$\sum_{i=1}^n f(i)$$ with respect to n.

That is, I have an function that is the sum of n terms and there is no closed form expression for it, and I need to get the derivative of the function w.r.t to n. I've tried to do it in Mathematica but just get error messages all the time, so is there any way to get the derivative w.r.t n of such a function?

It is possible, but you need to define the sum for non itegral n in some sensable way. I do not believe this can be done in all cases. In particular the extension should be as one expects. For example convergence and consistensy allowing take a function
$$g(n)= \sum_{i=1}^n f(i)$$
defined for n=0,1,2,3,4,...
and extend the definition to
$$g(n)= \sum_{i=1}^{\infty} [f(i)-f(i+n)]$$
Where we allow n to take new values
Then we may formally differentiate term by term to obtain
$$g'(n)= -\sum_{i=1}^{\infty} f'(i+n)$$
Which under favorable conditions (i.e. uniform convergence) we can justify