Intereting Posts

Does there exist a $1$-form $\alpha$ with $d\alpha = \omega$?
Examples of useful Category theory results?
Row swap changing sign of determinant
Combinatorial proofs: having a difficult time understanding how to write them out
SIB 2009, Problem #2
Big-O proof showing that t(n) is O(1)
Find $\sum_{i\in\mathbb{N}}(n-2i)^k\binom{n}{2i+1}$
Pivoting and Simplex Algorithm
Finding $\lim_{x\to \pm\infty}f(x)$ where $a,b>0$
Rotate a point in circle about an angle
Verifying proof :an Ideal $P$ is prime Ideal if $R/P$ is an integral domain.
Trigonometric and exp limit
Using the Determinant to verify Linear Independence, Span and Basis
$AB-BA=A$ implies $A$ is singular and $A$ is nilpotent.
The set of all finite sequences of members of a countable set is also countable

I noticed something odd while messing around on my calculator.

$$\lim_{x\to \infty} \cos^x(c)=0.7390851332$$ Where $c$ is a real constant.

My calculator is in radians and I got this number by simply taking the cosine of many numbers over and over again. No matter what number I use I always end up with that number. Why does this happen and where does this number come from?

- Is there a better counter-example? (problem involving limit of composition of functions)
- Two Limits Equal - Proof that $\lim_{n\to\infty }a_n=L$ implies $\lim_{n\to\infty }\frac{\sum_1^na_k}n=L$
- Proving a scalar function is differentiable at the origin but that its partial derivatives are not continuous at that point.
- Limit $c^n n!/n^n$ as $n$ goes to infinity
- Help with $\lim_{x\rightarrow +\infty} (x^2 - \sqrt{x^4 - x^2 + 1})$
- Showing that $ \displaystyle \lim_{n \rightarrow \infty} \left( 1 + \frac{r}{n} \right)^{n} = e^{r} $.

- Prove that $\lim\limits_{(x,y) \to (0,0)} \frac{{x{y^2}}}{{{x^2} + {y^4}}} = 0$
- How to prove that $\cos\theta$ is even without using unit circle?
- How is it solved: $\sin(x) + \sin(3x) + \sin(5x) +\dotsb + \sin(2n - 1)x =$
- Limit, solution in unusual way
- Is $f(x)=10$ a periodic function?
- 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.
- Solve a seemingly simple limit $\lim_{n\to\infty}\left(\frac{n-2}n\right)^{n^2}$
- Find $\lim_{n \to \infty} n^2 \sum^n_{k=1} {1 \over {(n^2+k^2)^2}}$ using Riemann sums
- Convergence, Integrals, and Limits question
- Show that $\cos (\sin \theta)>\sin (\cos \theta)$

What you have found is the unique, attractive fixed point of $\cos(x)$.

For more on this point and these terms, see this (MathWorld) and this (Wikipedia).

This is the unique real solution $r$ of $\cos(x) = x$.

For any $x \ne r$ we have $|\cos(x) – r| = \left|\int_{r}^x \sin(t)\ dt\right| < |x – r|$.

This implies that $r$ is a global attractor for this iteration.

As already discussed in other threads:

What is the solution of cos(x)=x?

Solving $2x – \sin 2x = \pi/2$ for $0 < x < \pi/2$

fhe fixed point of $\cos(x)$ (i.e. the Dottie number) can be written as a particular solution of Kepler equation, therefore it can be also expressed as:

$$ DottieNumber=\sum_{n=1}^\infty \frac{2J_n(n)}{n} \sin\left(\frac{\pi n}2\right)= 2\sum_{n=0}^\infty \left( \frac{J_{4n+1}(4n+1)}{4n+1} – \frac{J_{4n+3}(4n+3)}{4n+3}\right)$$

where $J_n(x)$ are Bessel functions.

its the solution to cos(x)=x, also sometimes known as the dottie number

The number is

$$\alpha=\frac1\pi \int_0^{\pi } \arctan\left(\tan \left(\frac{t-\sin t+\frac{\pi }{2}}2\right)\right) \, dt+\frac{1}{\pi }$$

You may treat is as dynamical system with state transition function $f(x) = cos(x)$. After first two iterations $f^{n > 2}(x)$ will lay in interval $I = [cos(1), 1]$. Line $g(x) = x$ will intercept $cos(x)$ in interval $I$ exactly once so $cos(x)$ has unique fixed point in $I$. Because of unique fixed point and because $|f'(x)| < 1$ sequence $f^n(x)$ will converge to this fixed point.

You found the real solution of $ \cos x = x $ through a fixed point converging process.

- Covariance of order statistics (uniform case)
- complex analysis(complex numbers) – Apollonius circle (derivation of proof)
- What are the quadratic extensions of $\mathbb{Q}_2$?
- $|a|=m,\,\gcd(m,n)=1 \implies a$ is an $n$'th power
- How to calculate the expected value when betting on which pairings will be selected
- Is mathematics one big tautology?
- $\int_0^{2\pi}e^{a \cos{\theta}}\cos({\sin{\theta}})\,d\theta$ using residues
- Is there a systematic way of solving cubic equations?
- Evaluate $ \int_{25\pi/4}^{53\pi/4}\frac{1}{(1+2^{\sin x})(1+2^{\cos x})}dx $
- Need a counter example for cycle in a graph
- Why is an empty set not a terminal object in categories $\mathsf{Top}$ and $\mathsf{Sets}$?
- Inverse of a Function exists iff Function is bijective
- If $\forall x \exists y : R(x, y)$, then is it true that $y = y(x)$?
- How do we know that $x^2 + \frac{1}{x^2}$ is greater or equal to $2$?
- Comb space has no simply connected cover