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 Curious
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Jun
30
comment Integral of exponencial
I was wondering if you considered applying residue calculus? mathfaculty.fullerton.edu/mathews/c2003/ResidueCalcMod.html
May
12
comment What is the strongest possible statement of the idea that “the tangent line is the best linear approximation”?
How do you define "best"? Anything close to "best fitting curve" technique? E.g. web.iitd.ac.in/~pmvs/courses/mel705/curvefitting.pdf
May
6
comment Intuition behind Khinchin's constant
Just a quote from en.wikipedia.org/wiki/Khinchin%27s_constant "... it can be proved that $T$ is an ergodic transformation of the measurable space $I$ endowed with the probability measure $\mu $ (this is the hard part of the proof). The ergodic theorem then says that for any $\mu $-integrable function $f$ on $I$, the average value of $f \left( T^{k} x \right)$ is the same for almost all $x$ ..."
May
2
answered Check the convergence of $a_{n+1}=\sqrt{a_n+\frac{4}{a_n}}$ where $a_1=4$
Apr
29
answered Guessing number in set 1-100 with weighted questions.
Feb
26
accepted What type of number is x?
Feb
26
accepted Prove that for any give sequence of digits, there is a perfect square starting with that sequence
Jan
29
answered how to compute the last 2 digits of 3^3^3^3 to n times?
Jan
6
answered Can a coin with an unknown bias be treated as fair?
Oct
22
comment It is easy to show that $S_m=\sum_{n=1}^\infty \frac{n}{2^n + m}$ converges for any natural$\ m$, but what is its value?
Not sure this is going to help (didn't try and don't have time for this), but did you try to find an asymptotic form? I typically start with $f(x)=\frac{x}{2^{x} + m}$ and $f(x)$ is descending from some $x$. Then $S_{m}=\sum_{n=1}^{\infty }f(n)\cdot (n+1 - n)\approx \int_{1}^{\infty }f(x)dx$. Then check this wolframalpha.com/input/?i=x+%2F+%282%5Ex+%2B+m%29
Sep
6
comment Is the sequence $(a_n)$ defined by $a_n=\tan{a_{n-1}}$, $a_0=1$, dense in $\Bbb{R}$?
Yes, it is. Consider $\tan(x)$ on $[ 0,\pi )$ (not $[-\frac{\pi }{2},\frac{\pi }{2}]$). Still, $\forall \alpha \in \mathbb{R}, \exists \beta \in[0, \pi): \tan(\beta)=\alpha$. If $\{b_{n}\}_{n=0}^{\infty}$ is dense in $[ 0,\pi )$, then $\exists b_{k}\approx \beta$ and $\alpha=\tan(\beta)\approx \tan(b_{k})=a_{k+1}$ (because $\tan(x)$ is continuous, except $\frac{\pi }{2}$, but it's easy to deal with it). And $\forall \alpha \in \mathbb{R}, \exists a_{k+1}: \alpha \approx a_{k+1}$, which means $\{a_{n}\}_{n=0}^{\infty}$ is dense in $\mathbb{R}$.
Sep
5
comment Is the sequence $(a_n)$ defined by $a_n=\tan{a_{n-1}}$, $a_0=1$, dense in $\Bbb{R}$?
Another trick that might turn to be useful, just narrowing down the problem. Consider $k_{n}\in \mathbb{Z}: k_{n}\cdot \pi \leq a_{n}<(k_{n}+1)\cdot \pi$ and the sequence $b_{n}=a_{n}-k_{n}\cdot \pi \in [0,\pi)$. We have $\tan(b_{n})=\tan(a_{n})=a_{n+1}$ or $\{\tan(b_{n})\}_{n=0}^{\infty}\bigcup \{1\}=\{ a_{n}\}_{n=0}^{\infty}$. If we "shift" $\{b_{n}\}_{n=0}^{\infty}$ by $t\cdot \pi, t\in \mathbb{Z}$ we get the same result. As a result, it's enough to prove that $\{b_{n}\}_{n=0}^{\infty}$ is dense in $[0,\pi)$
Aug
28
comment Is the sequence $(a_n)$ defined by $a_n=\tan{a_{n-1}}$, $a_0=1$, dense in $\Bbb{R}$?
Another result that might turn to be useful math.ust.hk/~majhu/Math203/Rudin/Homework15.pdf, point 4.4. Basically $\left \{ a_{n} \right \} = \left \{ \tan\left ( a_{n} \right ) \right \}\bigcup \left \{ 1 \right \}$
Aug
28
comment Is the sequence $(a_n)$ defined by $a_n=\tan{a_{n-1}}$, $a_0=1$, dense in $\Bbb{R}$?
Assume a function $f:A\rightarrow B$ that is surjective, continuous and periodic with period $T\in \mathbb{R}\setminus \mathbb{Q}$. Because $T$ is irrational, according to mathworld.wolfram.com/KroneckersApproximationTheorem.html, $\left \{ k\cdot T+n |n,k\in \mathbb{Z} \right \}$ is dense in $\mathbb{R}$, so $\forall \beta\in B,\exists \alpha \in A : \beta =f\left ( \alpha \right )\approx f\left ( k\cdot T+n \right )=f\left ( n \right )$ or $\left \{ f\left ( n \right ) |n\in \mathbb{Z}\right \}$ is dense in $B$, just a sketch ...
Aug
27
comment Is the sequence $(a_n)$ defined by $a_n=\tan{a_{n-1}}$, $a_0=1$, dense in $\Bbb{R}$?
I am wondering if the fact that $\left \{ \tan(n)| n\in \mathbb{Z} \right \}$ is dense on $\mathbb{R}$ could be of any use?
Jul
17
awarded  Curious
Jul
16
comment Prove that for any give sequence of digits, there is a perfect square starting with that sequence
From the educational perspective "so the solutions provided should include these tools (a bit of constraint to the problem)." ... I am sure there are other ways to prove it.
Jul
16
answered Prove that for any give sequence of digits, there is a perfect square starting with that sequence
Jul
16
asked Prove that for any give sequence of digits, there is a perfect square starting with that sequence
May
23
comment Given two concentric circles with radiuses r < R, can we estimate the number of chords in between the circles?
Another way, probably, to tackle the problem (and this is a brainstorming): $$n_{of chords}\sim \frac{\pi }{\sqrt{1-\alpha ^{2}}}=\pi \cdot \sqrt{1+\alpha ^{2}+\alpha ^{4}+...}< \pi \cdot \sqrt{n}\cdot \sqrt{1+\frac{1}{n}\sum_{i=n}^{\infty }\alpha ^{2\cdot i}}$$