I always thought there's some special reason, something behind the scenes that mathematicians used to construct those series. But, it turns out it may not be so. Or, whatever reason they had, it may not be a mathematical reason. The answer is, anyone can declare a series! By ZFC axioms, you are ALLOWED to form a series without any proof, almost directly from fundamental axioms of mathematics! You don't need to wait for high school education to do that. Once you are familiar with fractions, go ahead, create, construct, your own number series. And right there ask your teacher to introduce you to that delta-espilon definition of limit! No proof needed for constructing a series! How so? Because, essentially, fundamental axioms in math tell you that you can create numbers at free will, you can define sequences as you wish. So, you CAN define, say number 2, then multiply by itself, 2, 3, 4, 5,.. times and each time you decide to divide 1 by that and then sum them. Like here, 1/2 + 1/(2^2) + 1/(2^3) + 1/(2^4) + ..... Notice how arbitrary you came up with choosing the number and defining the sequence of operations. And you can define these sequences as much as you want. There is no other way! Nothing else behind the scene.
Now, proving whether it converges or not is a different story. However, look that YOU can define WHAT needs to be proven! Why would you define any series? Mathematician will say because you can! Do you really want to? You may not want to! That's the state of affairs too. Now, once you define it, math does not care whether you liked it or not, were you motivated or not, or, whether the series came form some physical or other process measurements, or from you just playing with numbers,. The series, as it is, can come into existence just because you can create them. You are allowed, again, by fundamental axioms in math to do that. And there can be many, many series, but again, they are all arbitrarily created. Try it by yourself! Just create a series, a sequence and try to find the sum. That's all what's to it.
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