If you take math in your first year of college, they teach you about infinite series and the radius of convergence. An infinite series is a polynomial in a variable X with and infinite number of terms ...
Adding up an infinite number of numbers doesn't always give a sensible answer. Suppose all the Ki are one, but X=2: we then get the sum 1 + 2 + 4 + 8 + 16 ... which clearly becomes infinite. (Sometimes we say the sum converges to infinity, but usually we say it diverges). If we set X=-2, we get an even worse problem: 1 - 2 + 4 - 8 + 16 ... If we start on the left and add up the numbers in order, we get 1, then -1, then 3, then -5, then 11 ... we flip-flop back and forth between larger and larger positive and negative numbers.
If you work at it, you can convince yourself that for X bigger than one, or for X smaller than minus one, the infinite sum doesn't make sense (it diverges). For X smaller than one and bigger than minus one, the sum can be done. The radius of convergence for this function is one.
Four more things we should mention.
- It's called a radius of convergence because mathematicians like to plug in complex numbers, like X = U + i V, where i * i = -1. They can show that the series converges inside a circle U2 + V2 = R2, and diverges outside the circle.
- If you're clever, you can figure out that our sum (with all the Ki=1), when it converges, always gives 1/(1-X). An infinite series represents a function. (Taylor's theorm tells us how to get the series from the function.) Other infinite series Ki can be used to make other functions.
- This sum 1/(1-X) does become big when X gets close to X=1. On the other hand, there doesn't seem to be anything wrong at X=-1. The convergence of the infinite series at X=-1 is spoiled because of a problem far away at X=1, which happens to be at the same distance from zero! The radius of convergence is usually the distance to the nearest point where the function blows up or gets weird.
- There is a simple way to calculate the radius of convergence of a series Ki (the ratio test). The series can't possibly converge unless the terms eventually get smaller and smaller. If we insist that |Kn+1 Xn+1| be smaller than |Kn Xn|, that can happen only if |X| is smaller than |Kn/Kn+1|. The radius of convergence of an infinite series is (basically) the value of |Kn/Kn+1| for large n.