geometric series for fractional n I have the following question. Consider the summation
$$
C/(1+r)^i
$$
which is for i = 1 to n, but n can be FRACTIONAL. Is there a mechanism to do that? Do we proceed like if it was anormal sum?
 A: Your sum is a geometric sum :
$$
\sum_{i=1}^n \frac{C}{(1+r)^i} = C \sum_{i=1}^n \left( \frac 1{(1+r)} \right)^i = C \frac{(1+r)^{n+1} - 1}{1+r - 1} = C \frac{(1+r)^{n+1} - 1}r.
$$
If you want to substitue fractional $n$ here, you can, the problem is the following ; you have a function that is currently defined over the positive integers (because I believe you said you are in a financial math context, so that you don't want people to give you -2 payments or something weird like that), and you want to extend it over $\mathbb Q$ (the rationals). There exists wayyy more than one way to do that, and perhaps the easiest way to do it is to plug in the $n$ as a fraction in your formula, but maybe it is not the most natural way : maybe there is a function defined over $\mathbb Q$ that modelizes your financial context better, but is only equal to your expression above when $n$ is an integer, and is worth something else when $n$ is not an integer. 
Formulas are not magical ; it's not because they work that you can play with them and expect them to do everything. The reason why they work is because there is a reason behind it, and you must find a reason behind it before expecting it to work all the time. (You don't need reason to notice that it works very often and then believe that it works all the time, but you need proof to be sure of it.) The whole point of research is to find those remarks/explanations.
A good question for you would be this : could you explain why it would make sense to consider a fractional $n$? Perhaps that with an answer to this question, one could give you a better formula to work with.
Hope that helps,
A: Sums by definition cannot have a non-integral number of terms.
However, in this case you can find closed form of the sum for integer $n$ using the standard trick for geometric series, and that closed form happens to be defined for fractional $n$. The result is, of course, not a "sum" in any principled sense, but it may (or may not) still make some sense to evaluate it, depending on what you're using it for. That depends completely on your application, though.
A: See
How to add a non-integer number of terms, and how to produce unusual infinite summations
Markus Müllera, Dierk Schleicherb
Journal of Computational and Applied Mathematics 178 (2005) 347 – 360
