Intereting Posts

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Good closed form approximation for iterates of $x^2+(1-x^2)x$
Every finite group of square-free order is soluble
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Proofs for an equality
Wedge product and cross product – any difference?
Is it possible to solve the Zebra Puzzle/Einstein's Riddle using pure math?
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Given a pair of continuous functions from a topological space to an ordered set, how to prove that this set is closed?
$\|(g\widehat{(f|f|^{2})})^{\vee}\|_{L^{2}} \leq C \|f\|_{L^{2}}^{r} \|(g\hat{f})^{\vee}\|_{L^{2}}$ for some $r\geq 1$?
Do the real numbers and the complex numbers have the same cardinality?
How to find the order of a group generated by two elements?
$C_{c}(X)$ is complete. then implies that $X$ is compact.

it can be demonstrated by elementary means that the curves $y=\cos x$ and $y=x$ meet exactly once, at a value $x=\alpha$ satisfying:

$$\cos \alpha = \alpha$$

it is also evident (empirically) that simple reiterated pressing of the cosine button on a calculator produces a sequence that seems to converge at a steady, modest pace, from any initial real value to $0.73908\dots$.

however for some time I have fiddled about ineffectually attempting to prove this convergence. this is interesting from a psychological point of view – because I believed the problem was just beyond my reach, I failed to spot a fairly simple proof-idea, which requires no more than high-school calculus and trigonometry.

or at least that is my present thought! the following proof-idea seems OK, though i haven’t dotted every *i* and crossed every *t*, and since I know myself to be error-prone, I would appreciate it if someone more experienced could check the argument, and remedy any deficiencies – or at worst detect some fundamental flaw I haven’t noticed.

- How to find the partial sum of a given series?
- Show that series converge or diverge
- If the $n^{th}$ partial sum of a series $\sum a_n$ is $S_n = \frac{n-1}{n+1}$, find $a_n$ and the sum.
- Is series $\displaystyle\sum^{\infty}_{n=1}\frac{\cos(nx)}{n^\alpha}$, for $\alpha>0$, convergent?
- Convergence of $\sum \frac{\sqrt{a_n}}{n^p}$
- Integral of odd function doesn't converge?

I am uncertain how to finish off, and also about how to properly manage the role of $\delta$ – in fact the convergence, though not rapid, seems very robust with regard to initial values.

thank you

firstly, since we know the number $\alpha$ exists, this simplifies the demonstration. it suffices to show that:

$$\exists \delta,\lambda \in (0,1) . \forall x \in \mathbb{R}.|x-\alpha| \lt \delta \rightarrow |\cos x – \alpha| \lt \lambda |x-\alpha|$$

set $\beta=\sin \alpha = \sqrt{1-\alpha^2}$ and let $x=\alpha +\epsilon$. then:

$$

|x-\alpha| = |\epsilon|

$$

and

$$\cos x = \cos \alpha \cos \epsilon – \sin \alpha \sin \epsilon = \alpha (1+O(\epsilon^2)) – \beta(\epsilon + O(\epsilon^3))

$$so:

$$ |\cos x – \alpha| = \beta |\epsilon|+O(\epsilon^2)= (\beta +O(\epsilon))|\epsilon|

$$ giving

$$ |\cos x – \alpha| = (\beta +O(\epsilon))|x-\alpha|

$$

- Convergence of Series
- System of equations, limit points
- Show that this limit is positive,
- Error Analysis and Modes of Convergences
- Power Series with the coefficients $n!/(n^n)$
- If $\sum a_n$ is a convergent series with $S = \lim s_n$, where $s_n$ is the nth partial sum, then $\lim_{n \to \infty} \frac{s_1+…+s_n}{n} = S$
- $f_n$ uniformly converge to $f$ and $g_n$ uniformly converge to $g$ then $f_n \cdot g_n$ uniformly converge to $f\cdot g$
- Convergence in measure of products
- Counter example to theorem in complex domain
- Evaluate $ \sum\limits_{n=1}^{\infty}\frac{n}{n^{4}+n^{2}+1}$

Here is a formal proof:

Let $f(x) = \cos x$ then the iteration is

$$x_{n+1} = f \left( x_n\right)$$

In the interval $[0,1]$, $\cos$ is decreasing and $ 0 \lt \cos 1$. Hence the function maps $[0,1]$ into itself. So by Brower’s fixed point theorem, $x_n$ converges to a fixed point.

Also in the interval $|f’| < 1$, so the mapping is a contraction eventually converges as $\rho^n$ where $\rho=\sin 1 \approx 0.84$

The slow convergence is due to the fact that $\rho \approx 1$

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- Entire function. Prove that $f(\bar{z})=\overline{f(z)}, \forall z\in C$
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- How prove this inequality $\frac{a+\sqrt{ab}+\sqrt{abc}+\sqrt{abcd}}{4}\le\sqrt{\frac{a(a+b)(a+b+c)(a+b+c+d)}{24}}$
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- Does the opposite of the brachistochrone exist?
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- How to prove Campanato space is a Banach space
- How to evaluate $I=\int\limits_0^{\pi/2}\frac{x\log{\sin{(x)}}}{\sin(x)}\,dx$
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