What is the significance of Convergence/Divergence of Series

Question

There are many ways and mathematical methods to test the convergence or Divergence of a series. What does Convergence or divergence indicate practically in the real world? Why is the convergence of series important?

Answer 1

Convergent series goes to a finite specific value so the more terms we add the closer to this we get. Divergent series on the other hand does not, they either grow indefinitaly in some direction or oscillate, as such the addition of more terms will cause it change value drastically.

This is why it's important also because when we calculate things we want to be able to get any degree of accuracy we seek which is difficult with divergent series as anything done there might make it blow up.

Answer 2

Convergence of series describes the idea of adding up countable infinite many values rigorously. For example, in probability, we have countable infinite many events $A_n$, we know the probability that event $A_n$ happens is $x_n=P(A_n)$, we want to know what is the probability that at least one of these events happens. How to compute it? Naturally, we ask whether it can be computed as the summation of the probability of each event (which means $\sum_{n=1}^\infty P(A_n)$)?

So in above question, there are two things, first, what does this infinity sum mean (which means whether the series converges)? Second, if it converges, it is equal to the value we want (what's the value the the series converges)?

Also in practice, we can only compute finite many terms, so is computer. If you know a series is convergent, and we want to study the value which the series converges to. Then we know if we choose large enough $N$, the finite sum $\sum_{n=1}^N x_n$ will be very close to $\sum_{n=1}^\infty x_n$, which can be chosen as a good approximation as the limit. This idea is very useful in computational science, or even mathematics itself.

Answer 3

When you punch in $2$ and $\sqrt x$ on a calculator (I hope that's practical enough) it is most likely that the calculator computes a partial sum $x_N$ of the series \begin{equation} x:=\sum_{n=1}^\infty r_n =\lim_{n\to\infty}x_n \text{ where } x_n:=\sum_{k=0}^n r_k ,\: r_0:=1 , \text{ and } r_n:=\frac12\Big(\frac 2{x_{n-1}}-x_{n-1}\Big). \end{equation} The fact that this series converges (and pretty fast) makes it an effective method for practical computations of square roots. Numerical analysis is the branch of mathematics that studies the practical meaning of convergence of real numbers.