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?

• How do you even define it? You must give a definition for $$\sum_{i=1}^n \frac{C}{(1+r)^i}$$ for $n \in \mathbb Q$ if you want to work with it. In the finite case, the sum is not a construction but just a concise notation. If that is your question, then I'd like to see if you have some motivation for this, if it is not just a wonder you're having. Those kind of things, when useful, are fun to work with. =) – Patrick Da Silva May 10 '12 at 19:34
• One option would be to take the formula for the sum of the series and extend it to non-integer $n$. – Jim Belk May 10 '12 at 19:34
• How to do that? That is precisely my question – Bober02 May 10 '12 at 19:35
• @Patrick OK, what you defined is the formula for annuity. it has a solution in the form A/r(1 - 1/(1+r)^n). Could I simply substitute fractional n now? – Bober02 May 10 '12 at 19:38
• Are you asking about your ability to substitute in a fractional $n$? I'm sure you're able to. Whether the result will make any sense cannot be answered without knowing what you're going to use it for. – Henning Makholm May 10 '12 at 19:42

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,

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.

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