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 Aug 29 comment What is the sum of this series? $\sum_{n=1}^\infty\frac{2^n (-1)^{n+1} B_n}{n}$ @Daniel Fischer well I would be OK with any of the summation methods, such as mean vanues etc. Aug 29 comment What is the sum of this series? $\sum_{n=1}^\infty\frac{2^n (-1)^{n+1} B_n}{n}$ @michaelrccurtis Bernoulli numbers. Aug 23 comment What happens if to introduce infinite and infinitesimel quantities this way? "would seem to contradict the definition" - it would not if $\varepsilon^2=0$ as in dual/parabolic numbers. The given definition as I figured out leads exactly to the dual numbers system. Aug 23 comment What happens if to introduce infinite and infinitesimel quantities this way? Thank you for pointing out Euler's work. It seems this definition leads exactly to the system of dual numbers: en.wikipedia.org/wiki/Dual_number In dual numbers we have exactly $\exp(\varepsilon)=1+\varepsilon$ although the first form of the definition does not work because division by $\varepsilon$ is undefined. Aug 23 comment What happens if to introduce infinite and infinitesimel quantities this way? @Hurkyl well I think this question can be closed because in dual numbers en.wikipedia.org/wiki/Dual_number we have exactly that: $\exp(\varepsilon)=1+\varepsilon$ Aug 23 comment What happens if to introduce infinite and infinitesimel quantities this way? @Hurkyl what if to consider $\varepsilon$ not just an infinitesimal but as just some new algebraic element with this property? Aug 23 comment Conventions adopted for extended reals @mercio no, we get 6=5. Aug 23 comment Conventions adopted for extended reals @mercio wrong. if we define 0/0=0, and devide the both parts of $0\cdot 17=0$ by 0, we get 0=0, no contradiction. Aug 23 comment Conventions adopted for extended reals @mercio what contradictions appear if we define $0/0=0$? Aug 23 comment What happens if to introduce infinite and infinitesimel quantities this way? @Hurkyl literally equal. Aug 23 comment Is there a name for complex numbers over affinely extended reals? @Hurkyl agreed, but what's the usual name (or symbol) for such set? Aug 23 comment Is there a name for complex numbers over affinely extended reals? @Mariano Suárez-Alvarez And inverse Mellin tranform uses the both in one formula: upload.wikimedia.org/math/5/4/a/… Aug 23 comment Is there a name for complex numbers over affinely extended reals? @Mariano Suárez-Alvarez Look for instance at the Wiki article about Forier transform. It uses both complex numbers and signed infinities a lot: en.wikipedia.org/wiki/Fourier_transform Aug 23 comment Motivation for different mathematics foundations ZC? Did you mean ZF? Aug 22 comment Is there a formulaic way to go from $\sum_{k=1}^{n} \frac{1}{k}$ back to $n$? Why is the downvote? Aug 21 comment Zero to the zero power - is $0^0=1$? The second "general rule" is wrong. It is correct only for positive numbers. Aug 13 comment Why the number of all reals is $2^{\aleph_0}$ and not ${\aleph_0}^{\aleph_0}$? But I think in this system one can define "sub-$\Omega$" numbers which would play the role of natural numbers Aug 13 comment Why the number of all reals is $2^{\aleph_0}$ and not ${\aleph_0}^{\aleph_0}$? @Asaf Karagila The smallest number which can be written using such "decimal" expansion using positions and base up to $\Omega$ will be, well... $\frac{1}{{(\Omega+1)}^{\Omega}-1}$. It is not real though indeed. Aug 13 comment Why the number of all reals is $2^{\aleph_0}$ and not ${\aleph_0}^{\aleph_0}$? @Asaf Karagila I am actually talking about a theory where it is not based on bijections. I have just applied brute force of a numerical sustem of infinite numbers (reals+infinites) so to find the quantity of reals and these are the results I obtained (using the notion of "smallest real" which apparently exists in such system). Aug 13 comment Why the number of all reals is $2^{\aleph_0}$ and not ${\aleph_0}^{\aleph_0}$? @Asaf Karagila I meant the system which is not invariant against addition