So typically for an additional relation $R$ which isn't trichotomous, It truly is obvious that "$alpha$ is infinite with regard to $R$" is just not reminiscent of "$alpha$ is transfinite with regard to $R$". Now the distinction between "infinite" and "transfinite" comes out.

$begingroup$ Inside your feedback, you requested how summing up finite points (or products and solutions of finite issues with other finite factors) can give you a thing infinite. This shouldn't be much too counterintuitive; The reasoning is that you're summing up infinitely a lot of Those people finite points. If anything at all, It truly is counterintuitive which you can occasionally sum up infinitely many things and have a finite consequence. I do concur, nonetheless, that the idea of a sport such as St. Petersburg Video game seems counterintuitive. Element of it would be because of the term "expectation." In typical utilization, when we be expecting anything to happen, we expect It is extra very likely to occur than not.

one $begingroup$ @MSIS: Take into account an infinite industry like $mathbb Q$. Just about every field is actually a Euclidean area. If there is a more elaborate set up for your difficulty, inquiring a brand new Concern (with that context) would be constructive. $endgroup$

with out applying Taylor sequence. extra explicitly without having employing calculus. how can We all know if a perform is usually expressed as series or not ?

$begingroup$ Primarily, you gave The solution your self: "infinity around infinity" isn't defined Simply because it ought to be the result of limiting processes of different character.

How can fighter jets compensate for Infinite Craft the curvature with the earth after they're traveling so minimal to the bottom?

Does there exist an infinite field with attribute $p$ for just about any prime $p$ that's not far too huge? ninety six

Some have noticed that you can generate the Taylor sequence for that at $r=0$. Yet another way is to utilize artificial division or polynomial extensive division. It can be hard to typeset in this article, but I am going to provide you with the flavor as ideal I'm able to.

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$begingroup$ As soon as you concentrate on probabilistic experiments with infinite outcomes, it is not hard to discover random variables with an infinite envisioned worth. Take into consideration the next illustration (that is only a game that yields an illustration similar to the one particular Yuri provided):

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