- calculus - What is infinity divided by infinity? - Mathematics Stack . . .
One advantage of approach (2) is that it allows one to discuss indeterminate forms in concrete fashion and distinguish several cases depending on the nature of numerator and denominator: infinitesimal, infinite, or appreciable finite, before discussing the technical notion of limit which tends to be confusing to beginners
- elementary set theory - What do finite, infinite, countable, not . . .
Clearly every finite set is countable, but also some infinite sets are countable Note that some places define countable as infinite and the above definition In such cases we say that finite sets are "at most countable"
- Can Hilberts grand hotel accommodate *infinite* layers of infinity?
These algorithms can be extended to further layers of nesting, e g an infinite number of ships, each containing an infinite number of coaches, each containing an infinite number of passengers (3 layers of infinity) However, my question is, does this apply to infinite layers of nesting?
- What is the difference between infinite and transfinite?
The reason being, especially in the non-standard analysis case, that "infinite number" is sort of awkward and can make people think about $\infty$ or infinite cardinals somehow, which may be giving the wrong impression But "transfinite number" sends, to me, a somewhat clearer message that there is a particular context in which the term takes
- Is the sum of all natural numbers $-\\frac{1}{12}$?
You are right to be suspicious We usually define an infinite sum by taking the limit of the partial sums So $$1+2+3+4+5+\dots $$ would be what we get as the limit of the partial sums $$1$$ $$1+2$$ $$1+2+3$$ and so on Now, it is clear that these partial sums grow without bound, so traditionally we say that the sum either doesn't exist or
- general topology - Show that the infinite intersection of nested non . . .
Claim: A topological space $\,X\,$ is compact iff it has the Finite Intersection Property (=FIP): Proof: (1) Suppose $\,X\,$ is compact and let $\,\{V_i\}\,$ be a
- Infinite class of closed sets whose union is not closed
Here is a good example which clearly shows that the infinite union of closed sets may not be closed
- calculus - Infinite Geometric Series Formula Derivation - Mathematics . . .
Infinite Geometric Series Formula Derivation Ask Question Asked 12 years, 3 months ago Modified 4 years
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