2

I am learning about convergence/divergence etc. I was just wondering, what is the difference? It seems a bit technical to me - especially once you start talking about absolute and conditional convergence. Are there practical ramifications of all this?

Burt
  • 1,851
  • Absolutely. Throughout math and science. – saulspatz Dec 27 '19 at 03:56
  • Pretty much all of science is written in terms of math. Here's one example: the central limit theorem (CLT) which is used everywhere by scientists analyzing data, and is also the theoretical backbone of a lot of statistical tools. The CLT is stated in terms of convergence in distribution, which defined in terms of a sequence of probabilities, which is a sequence of real numbers. – Thomas Winckelman Dec 27 '19 at 04:03
  • Also, you would (through no fault of your own) not know this from freshman calculus, but a lot of extremely important tools are defined in terms of sequences. To name a few: the exponential function, and exponentiation to an irrational real power. – Thomas Winckelman Dec 27 '19 at 04:04

1 Answers1

1

This previously asked question may be of assistance: Are questions of convergence important in real life?

TL;DR: real-world situations and models are often approximations of theoretical results and so we use convergences to make predictions in many fields from physics to engineering to statistics as accurately as we can or need for our purposes.

Hopefully that makes sense and/or gives you an avenue for research.

Soham Konar
  • 454
  • Who made up what is considered converging? Technically, $\frac1n$ gets smaller and smaller and kinda converges to zero - it just takes "too long". So, who decided that this is "too long"? – Burt Dec 29 '19 at 01:45
  • I think that if we prove that something is converging if the value is always increasing/decreasing toward the limit but it can be proven that value never goes past the limit. – Soham Konar Dec 29 '19 at 03:13
  • 1
    Sorry - should've been clearer. Who defined what makes a series considered to be converging? $\sum_{n=1}^{\infty}\frac1n$ seems to be that the summands are heading towards zero even though it will take a really really really long time. – Burt Dec 29 '19 at 04:15
  • I am not completely sure but I think these links will help you/us: https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit, https://math.stackexchange.com/questions/65667/how-to-prove-a-limit-exists-using-the-epsilon-delta-definition-of-a-limit, https://math.stackexchange.com/questions/1627069/can-we-assign-a-value-to-the-sum-of-the-reciprocals-of-the-natural-numbers – Soham Konar Dec 29 '19 at 04:22