I forgot the point out which i would want to Categorical the defition with no applying Taylor collection, as it works by using calculus, which which i don't want to contemplate in the meanwhile. $endgroup$
$begingroup$ Within your responses, you requested how summing up finite items (or items of finite issues with other finite factors) can provide you with one thing infinite. This should not be way too counterintuitive; The theory is that you're summing up infinitely many of These finite things. If anything at all, It really is counterintuitive which you can occasionally sum up infinitely many things and acquire a finite outcome. I do agree, even so, that the concept of a game such as St. Petersburg Video game looks counterintuitive. Section of it'd be as a result of phrase "expectation." In typical use, after we expect some thing to happen, we expect It truly is a lot more probably to occur than not.
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These ideas are defined employing an buying, the idea of working with an arbitrary relation $R$ in this similar definition is really a phase in an alien course. $endgroup$
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Be aware that somebody may perhaps determine "transfinite" similar to "Dedekind infinite" that's an abuse of terms in my view.
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$begingroup$ I fully grasp the definition of $e^x$ by limit. But I will not understand how to think of:
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(the principal exception I understand of may be the prolonged hyperreal line, that has quite a few infinite quantities obeying the 'regular' legislation of arithmetic, and a pair of extra quantities we phone $+infty$ and $-infty$ which have the largest magnitude of all infinite figures, and don't obey the 'usual' rules of arithmetic)
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