What is the relation between a summation and an integral ? This question is actually based on a previous question of mine here where I got two answers (one is based on summation notation) and the other is based on integral notation and I do not know yet which one to accept . So I would like to understand the connect between the two ?
relation between integral and summation
23
$\begingroup$
calculus
sequences-and-series
integration
-
1Wikipedia has [a nice paragraph or two](http://en.wikipedia.org/wiki/Integral#Introduction) as an introduction. – 2012-08-21
-
2There is a certain relation between integral and summation given by $$ \sum\limits_{i\in I}f(i) = \int\limits_{I}f(x)\#(\mathrm dx) $$ where $\#$ is the counting measure, and the function $f$ is e.g. non-negative. Such connection is also described in article mentioned by FrenzY DT. However, in the previous version of yours the integral and summation are used for, say, orthogonal purposes hence this connection plays almost no role there. – 2012-08-21
-
0@Ilya by orthogonal you mean "mutually independent " ? – 2012-08-21
-
2Well, not in the sense of the probability. I meant, that there I used an integral as a definition of the expectation, and Seyhmus used the sum there *inside* the expectation. Well, the linearity just follows from the fact that you can *swap* integral and the sum (i.e. these two operators *commute*) - but an extremely important fact that *the sum is a special case of an integral, and the integral is a limit of sums* is not used there. In fact, you might have there another operator instead of the sum, which also commutes with the integral - e.g. the limit operator $\lim$. – 2012-08-21