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2d
comment Find analytically the sequence of iterations $x_n$ for Newton's method applied to the function $f(x) = x^2$ with the starting point $x_0 = 1$.
@Shalid $\cdots = x_n-x_n/2 = x_n/2$. Now, if you compute a few terms, you oughtta spot a pattern fairly easily.
2d
comment Find a Continuous Function with Cantor Set Level Sets
@StellaBiderman There is definitely a countable but dense set of points in $[0,2/3]$ for which $\tau^{-1}(c)$ is a Cantor set. I expanded my answer substantially to explain this and presented a couple of examples as well.
2d
revised Find a Continuous Function with Cantor Set Level Sets
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2d
answered What is the condensation set of a fractal?
Apr
28
comment How to prove $\sum n/3^n$ converges without ratio test?
You can use the integral test, as you can integrate $f(x)=x3^{-x}$ (which is positive and decreasing for sufficiently large $x$) using integration by parts.
Apr
27
revised Open source lecture notes and textbooks
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Apr
22
answered Find a Continuous Function with Cantor Set Level Sets
Apr
12
revised Is this a valid definition of “self-similar fractal”?
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Apr
12
comment Is this a valid definition of “self-similar fractal”?
... Note also that the question at hand refers specifically to "self-similar fractal*, which I took to mean "self-similar set". Now that is a specific concept and my answer essentially says that "invariant set of an IFS* is exactly correct while "doesn't tile the plane" isn't really relevant.
Apr
12
comment Is this a valid definition of “self-similar fractal”?
@user1952009 I didn't meant to confuse you! My own feeling is that "fractal geometry" is a subject and is as loosely defined as, say, "linear algebra". We know what we are studying when we do linear algebra - linear transformations, matrices, and many related topics. Similarly, we know what we are studying when we do fractal geometry: Iterated function systems, Hausdorff measure, and many related topics. I think the reason people haven't been able to define "fractal" is because the topics in fractal geometry are equally applicable to simple and complicated sets.
Apr
12
revised Is this a valid definition of “self-similar fractal”?
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Apr
12
comment Is this a valid definition of “self-similar fractal”?
@user1952009 Yes. Even stronger - the maps must be strict similarity transformations. Note that the maps here are, in fact strict similarity contractions and, therefore, must have unique fixed points, though they might be a little tricky to see.
Apr
12
comment Is this a valid definition of “self-similar fractal”?
@PyRulez Yes, it is tricky. I hope that the edit helps.
Apr
12
revised Is this a valid definition of “self-similar fractal”?
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Apr
12
revised Is this a valid definition of “self-similar fractal”?
added 361 characters in body
Apr
12
awarded  Revival
Apr
12
answered What is the Hilbert curve's equation?!
Apr
10
answered Is this a valid definition of “self-similar fractal”?
Apr
10
comment Prove that the iteration of $\sin(x)$ goes to zero as $n$ goes to $\infty$
Hint: $|\sin(x)| < |x|$ for all non-zero $x\in\mathbb R$.
Apr
10
revised Prove that the iteration of $\sin(x)$ goes to zero as $n$ goes to $\infty$
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