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Doing math.


Apr
13
comment Why do some series converge and others diverge?
@Ethan: Because in the sum you add those numbers together. If you add numbers that are large then obviously the end result will be large. If you add negligent numbers together, then end result will stay small.
Apr
13
comment Why do some series converge and others diverge?
@Ethan: I mean that the sum is always less than $1$, for any finitely many terms that you sum together. By converging fast enough to zero I mean the following. By looking at the sequences $(\frac{1}{n})_{n=1}^{\infty}$ and $(\frac{1}{2^{n}})_{n=1}^{\infty}$ you will see that both of them converge to zero, but e.g. the third term in the first sequence is $\frac{1}{3}$, whereas in the second sequence it is $\frac{1}{8}$: much less. And for any given $n\in\mathbb{N}$‚ the other sequence will always have small values than the other. The terms go faster to zero so to say.
Apr
13
comment Why do some series converge and others diverge?
@Ethan: By a simple observation you can see that whenever you add finitely many of those terms $\frac{1}{2},\frac{1}{4},\frac{1}{8},...$ the sum is always bounded above by $1$. So the infinite sum must also be bounded above by $1$. The infinite sum is defined as the limit of the partial sums. they might be diverging or converging to a specific number, that all depends on the numbers that you're adding. Usually you can think that if the terms of the sum converge "fast enough" to zero, then the corresponding sum is finite.
Apr
11
revised Why do some series converge and others diverge?
added 146 characters in body
Apr
11
revised Why do some series converge and others diverge?
added 146 characters in body
Apr
11
answered Why do some series converge and others diverge?
Apr
10
comment Using Cantor's intersection theorem
@Mitsos: It's false in general for continuous functions as well. It's true for example, always when $f$ is injective. You have to argue with what you know about the sets $F_{n}$ relationship to each other.
Apr
10
revised how to use holders inequality to show lq is a subspace of lp
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Apr
10
answered how to use holders inequality to show lq is a subspace of lp
Apr
10
revised Homotopy on the unit circle
added 190 characters in body
Apr
10
comment Using Cantor's intersection theorem
Why can we conclude $\cap_{n}f(F_{n})=f(\cap_{n}F_{n})$? This is false in general.
Apr
10
answered Simply connectedness in $R^3$ with a spherical hole?
Apr
10
answered Homotopy on the unit circle
Apr
10
answered Is it possible to list $\mathbb{Q}$ so that the result set to be a monotonic sequence?
Apr
4
awarded  general-topology
Apr
3
comment Limit point compactness
Note that there are possibly subsets of $X$ that are neither open or closed. So "not closed" does not necessarily mean "open".
Mar
30
comment Does this limit give us the Cantor set?
When you say $\lim_{n\to a}x^{n}$, what exactly do you mean there? Under which metric are you limiting this sequence of sets? And what is $a$ in the limit?
Mar
19
comment Do open maps move isolated points to isolated points, in general?
Do you mean that $O$ is an open subset of $X$ instead of $A$? Because otherwise wouldn't we have $A\cap O=O$?
Mar
16
awarded  Nice Question
Mar
6
comment Using a contradiction to show something is not compact
@MPW: If you can take the continuity of determinant as obvious, then the set in question equals $\mathrm{det}^{-1}(\mathbb{R}\setminus\{0\})$ and is thus open. It should also be very elementary that $\mathbb{R}\setminus\{0\}$ is an open subset of $\mathbb{R}$.