Fifteen years and they noded some obsolete tech buzzword but NOBODY NODED THIS! OK, well, until someone spends several days addressing this subject thoroughly, this will just have to do.
Convergence is a concept at the heart of mathematical analysis. It's probably the single most important concept. It's the only way you can pin down the value of almost all real numbers; indeed, it's the only context under which an "actual" value exists for most of them.
Let's play a little game. We're going to chop up the number 2 and accumulate some of the pieces.
Start with A=0 and R=2.
For each integer N, we're going to calculate the value of F(N)=2*A*2-N+2-2N, and we'll make a note of whether or not it's greater than R. If it's too big, we'll just skip it. If it's not too big, we'll subtract F(N) from R and add 2-N to A.
F(0)=1 --> A=1; R=1
F(1)=1.25 (too much; skip)
F(2)=0.5625 --> A=1.25; R=0.4375
F(3)=0.328125 --> A=1.375; R=0.109375
F(4)=0.17578125 (too much; skip)
F(5)=0.0869140625 --> A=1.40625; R=0.0224609375
F(6)=0.044189453125 (too much; skip)
F(7)=0.02203369140625 --> A=1.4140625 R=0.00042724609375
You should be able to see that as N increases indefinitely, F(N) and R keep getting smaller and smaller. A can only change by at most R in each step, and so the difference between successive values of A (called partial sums) keep getting smaller and smaller, and closer and closer to each other. In this case, we can prove that the value of A will keep getting closer and closer to (but will never exceed) the square root of 2. As N gets large and larger and the difference between partial sums get smaller, some engineer or physicist may eventually decide she's had enough and stop, and use the current value of A for the "square root of 2".
But a mathematician has infinite patience and can go on forever. And that's the point: What happens if we go on forever? Do we get the "actual" square root of 2?
This was the dilemma facing late 18TH and early 19TH-century mathematicians. Two surpassing 17TH-century geniuses, Newton and Leibniz, had simultaneously invented calculus, a new way of squaring the circle and solving other problems that couldn't be solved by conventional algebra or geometry. Calculus, with its amazing predictive power revolutionized science and engineering, making the Industrial Revolution possible.
Unfortunately, a curmudgeonly Irish clergyman (no not that one, but his contemporary Bishop Berkeley) played spoilsport, showing the absurdity of flawed concepts built into both forms of calculus: For example, Differentials were thought to hold an "infinitesimal" value smaller than any finite value but larger than zero. Infinitesimals caused paradoxes right away in some of the most fundamental calculations.
The trouble was, calculus worked.
It was the job of later mathematicians to figure out why. And they eventually did: Cauchy worked out the concept of the limit of a convergent series. In his Cours d'Analyse he first used a notation that is used to this day, involving the two Greek letters δ (delta) and ε (epsilon).
Cauchy's concept of limit was less than completely rigorous, but Weierstrass was able to introduce concepts such as "uniform convergence" which finally allowed us to say
A function F(X) converges to a "limit" value L at a value X0 if for each real number δ there is a real number ε such that if |X-X0| < δ, then F(X)-L < ε
If we have such a limit L, we can usually say F(X0) = L.
We have a tool to assure ourselves that F(X0) has unique "actual" value (the limit L) if we can calculate the values of different F(X0+δ) and they keep getting closer and closer as δ gets smaller and smaller. We can sidestep infinitesimals, saying that after an infinite number of steps of our earlier calculation, R=0.
This answers the question we asked earlier on: Yes, if we are allowed to go on forever, we must obtain a "final" unique value of the square root of 2.
We can also define convergence in ways that don't involve real numbers: if we have
- a function F between two sets A and B (F:A->B)
- and a collection of nested subsets S in A with some a∈A in all of them,
- and if for each b∈A where b≠a there's a set s∈S where b∉s,
- defining F(s)={F(t)|t∈s},
then we can impose a structure on B with another such collection F(S) in B, so that there is a single element b∈B where F(a)=b. We are able to move on to the concept of the continuity of the function F based on the structure of B.