Divergence and convergence
“How can we test when a mathematical pattern will converge to a single number or diverge into infinity?”
Let me ask you a question. Suppose that I were to tell you that a country was completely made up of immortal individuals. And let’s say that these individuals were to reproduce their population at an ever changing rate. How could we tell if their population would expand into infinity, or stop at a certain number? Well, let’s solve a numerical example to figure this out.
Suppose that this same country starts out with one hundred people. And let’s have these people reproduce so that the first year, they would produce half of the population (50 in this case), and the next year, they would produce only half of a half of their initial population (25 ), and so on. Eventually, the number of the people in this country would converge into a single, static number! (200 for this example).
Now let’s the the same population but with a different rate. In this case, the rate of the change of population will be double rate of the prior time, so after the first year, the country will add 200 people, and the next year, the country will add 400 people, and so on, until the number of people in this country will diverge at infinity!
So let’s think about this some more. If the rate of reproduction decreases with each value, then it will eventually shrink down to zero, and if the rate of reproduction increases with each value, then it will expand into infinity.
Well, now let’s apply this mindset to a more abstract mathematical entity. Let’s take the infinite series n=0∞1xn, with the condition that we ourselves can choose that values for x!. Well, for all x values that are greater than one, the value will get smaller as time goes on, and eventually the number will converge onto a single number. Likewise, if our x values are smaller than one, then the rate will be increasing, resulting in an ever expanding number.