Mark S.
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 Sep 23 comment The numerical relation of the sum of two divergence series C. I would argue against with your statement 2. See the hyperreals section of my answer for a way of thinking that gives you a relevant quantity. D. In statement 3., be careful with set builder notation, as it denotes a set, not a sequence or series. The set of values of the sequence of terms is neither the sequence of terms, nor the sequence of partial sums. E. As with statement 2., I object to statement 3.'s "we cannot make meaningful statements about the sum of all elements in $S_1$" and "the word 'twice'...belongs to the domain of [the finite] alone" on similar grounds as my points A.,C. Sep 23 comment The numerical relation of the sum of two divergence series I think some of your wording is too general, or misleading (at least to all who are not studying for a Calculus exam right now). A. Your introduction implies that there are no ways to define two different quantitative infinities, but that is decidedly not true. Ordinals, Cardinals, and non-archimedean fields (and proper class versions thereof, like the Surreal numbers) are all options for doing so. B. I don't understand why you claim that all quantifiable infinities are in "cartesian co-ordinate space" that space (or why you think the OP is claiming that?). [Continued in next comment.] Sep 21 awarded Enlightened Sep 21 awarded Nice Answer Sep 17 comment The numerical relation of the sum of two divergence series The limit of the ratio of the partial sums being $\frac12$ is a relevant observation, perhaps the most relevant to the original question. Sep 16 comment The numerical relation of the sum of two divergence series @Jerry, IIRC, the operations in that video are sort of engineered carefully to get the right answer. Those sorts of operations don't always preserve all interpretations of sums. Sep 16 comment The numerical relation of the sum of two divergence series @martycohen, Comparing the graphs for a region of large x it's like the regularities you'd get when sampling $\sin x$. Asking about those regularities might warrant its on question if you can't find a satisfying answer by looking up introductions to sampling, etc. Sep 14 comment The numerical relation of the sum of two divergence series @iMath In short: By themselves, functions can't be treated quite like numbers: they don't have a nice order, you can't divide by every non-zero function, etc. Assuming you want "an infinity" to be a genuine number that you can add, divide by, multiply by 2 or another infinity, etc., then functions alone doesn't do that for you. You need a new idea to turn the vague idea of "functions except asymptotic ones should be nearly the same" into a full-fledged number system. Sep 12 answered The numerical relation of the sum of two divergence series Sep 10 comment The digital root of factorials from 6! to infinity! is always 9. The digital root of any number divisible by 9 is 9. Have you seen the divisibility test for 9? Sep 5 comment The set of all points $\in \mathbb{R}^3$ twice as far from $\mathbf{a}$ as they are from $\mathbf{b}$? @Rahul It's also 2-dimensional under a number of other interpretations, like "Hausdorff dimension". Sep 5 answered Plot a single point on number line in interval notation Sep 5 answered Trying to find $\lim_{x\to 0^+} \frac{x^2\sin(1/x)}{\sin x}$, I get $\frac{\infty}{0}$, what is that? Sep 5 comment Why is the slope-intercept form of the equation of a line often written $y=mx+b$? Why $m$ instead of $a$? I've seen a book or two claiming m comes from a french word related to "mount", but maybe that's not backed up by anything. Sep 1 comment If $\{x_i\}_{i=1}^n$ are the roots of $f(x)=a_nx^n + a_{n-1}x^{n-1} + \ldots +a_0$ then $\sum_{i=1}^nx_i^{n-1}$ is independent of $a_0$ Good observation! This follows immediately from Newton's Identities Aug 16 answered Proving the rules of a complicated game are well defined Aug 14 comment Volume of 3D shape with parallelogram as base It sounds like there are (differential?) equations about elastic materials that could help pin down the shape, but even if you asked on another site where people are more likely to know the relevant equations, you'd still need to give data about the initial conditions (where are the fibers that dont move, what are the constants related to how elastic the material is,etc.). Alternatively, if you have the code that generated that image, perhaps it could be reverse engineered and/or the volume approximated with a Monte Carlo method. Aug 14 comment Volume of 3D shape with parallelogram as base How are the bulges defined? They don't look like pieces of a sphere. Do you have a parametric formula for the shape? What error would you allow? Etc. Aug 14 comment Volume of 3D shape with parallelogram as base It depends what shape it is (the picture looks close to half of the intersection of two cylinders, but you said parallelogram so we don't know how your shape differs) and what you know (are you comfortable with multivariate calculus already?). Aug 14 comment If all derivatives are zero at a point, what does this imply? It need not be analytic (if it were, it would be constant). Look up bump functions. If you hadn't said "differentiable nearby"n I would say it need not be continuous at any other point:combine a bump function with an example of differentiable at a point but discontinuous elsewhere.