1 $begingroup$ @lhf: I under no circumstances miss a chance to attract the "infinitude of primes". $endgroup$
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But can it be probable to precise the summation definition of $e^x$, without utilizing them ? Considering the fact that, I am regenerating my math expertise I desire to go comprehensive to calculus, differential equations and so on. $endgroup$
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From my perspective, the infinite query hasn't been solved. We even now Never understand what infinite essentially is, Though Now we have described a lot of its Qualities pretty much. It's possible, if we start out exploring outside the boundaries from the stablished axioms we may reach some conclutions...
For example, when you substitute "$infty$" in my solution with "a little something infinite", you obtain something that is smart! Which of course ensures that you assumed "$infty$" to become a suitable noun, not me! =P $endgroup$
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These decisions/conventions must be taken in this kind of way that The foundations of multiplication (e.g. $xsituations y=ymoments x$) continue to be valid as much as possible. Quite a position! Your instinct claims that for $(2,infty)$ it is an effective factor to choose $infty$ as products. That confirms to me that your intuition should be to be highly regarded. And recall: intuition is essential in mathematics!
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For example, the list of all purely natural numbers $mathbb N$ is "infinite" in cardinality, actually "countably infinite" -- but its cardinal $aleph_0$ and also the ordinal $omega$ that is the "order style" of $mathbb N$ are described as staying "transfinite".
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As for the issue about irrespective of whether a function is often expressed being a sequence or not, to reply it I think you need to say anything about calculus. What I indicate is that if a "great" purpose $f(x)$ provides a series representation at a point $a$ then the sequence is presented by