MATH: Infinite sums and interesting results

By Bruno;

Hey everyone! It has been a while since my last post, but now I am back! Recently, I have studied lots of things on infinite sums, and if you don't know what an infinite sum is, It is a sum of infinitely many terms, such as 1+2+3+...; This is the basics, and as I'm running out of time (in the moment I am writing this post), I think it is interesting to define some functions here to show the 'interesting results' highlighted in the title.

Consider the following function in the variable x

It is well-known that for x>1, it always converge(Here, converge means that this function assumes a finite real value ). I mean, it is really funny, right? And the best thing comes now: there are lots of integer values of x that this function is a power of Pi(3,1415...) times a rational number! Here you are some values of x that satisfies it:

Actually, there is a a formula when x=2n, with n integer. However, I don't know any known result about odd values of x.

As you saw, the notion of convergence is extremely important for math, for instance, it is the basis of what I've just introduced you. But around this wonderful world of numbers, some results are even more mind-blowing than this Pi related stuff. Actually, if you change the order of all the terms in the infinite sum, you can vary its value. That is really impressive! And the first time that I was introduced to it was during a math Olympiad test in 2017. One example is ln 2 and (ln 2)/2, where ln 2 is the natural log of 2.

Note that we just inserted some parenthesis and changed the order of the terms, but as the sum is infinite, it also changed the value of the function. Indeed, very impressive!

Well, It was quite a short post, but enough, I guess. As we say in my home town, 'PAPA IS ON', which means 'I am back'.