Keep in mind if we discard x/x or rather just x, then all the complexity is discarded with it be discrete or even infinite in which case entirely new computations can not only be optimized but simply made possible to achieve with our limited hardware.

To put it another way, the 'magic' of Calculus is nothing but the Sigma operation producing a pattern like (x/x)*someEquation combined with the fact that something (such as (x/x)) divided by itself equals 1, and the fact that 1 multiplied by 'someEquation' causes 'someEquation' no change, which means large parts of the equation are not needed to compute the Sigma operation at some iteration. Finding these 'derivatives' is a huge part of both mathematics and algorithm development. You want to know the 'magic' of Derivatives? It's hard to explain, but basically, sometimes, just sometimes when N = Infinity or some other value, and again just SOMETIMES, if the algebra lines up in such a way so that the algebra produced cancels itself out, we are left with a new simplified algebra which no longer takes 'infinite' or 'N iterations' to reach some wanted result. If you want to be pedantic (and you should) you might also say that the Limit has two parts, i's original value at the 'start' of 'execution' (Lambda Calculus does not relate computation to neither time nor a machine, but you can always strip that model away once you get the concept) is the 'lower limit' and i's 'final value' is the 'upper limit.' In English, if N is 100 then the Limit would be 'where i=100.' If N=5 then the 'Limit' is 'where i=5' and so on. What is the 'Limit?' Well in C you would define it as (i=1 i<=N ). Given the number on the right hand side of Sigma we are told this sum starts at 1, soДЄll the way up until the LIMIT where you will have Anyway, lets just plug and chug through the 'loop'.

$sigma math$

Again I'm no guru, so holler if you spot mistakes. I'm by no means a calculus guru, just a layman, but perhaps the layman's terms will be useful for somebody.

Sigma math