because...
\begin{align}
C_0(1)_1 & =\{j,\omega[i]j,...,{}^2\omega,...,\epsilon_0[\omega_1]|j\in\Bbb{N},i=\{1,2,3\}\},\eta=0\\
C_0(1)_2 & =\{j,\omega[i]j,...,{}^3\omega,...,{}^2\epsilon_0,[\omega_1]|j\in\Bbb{N},i=\{1,2,3\}\},\eta=0\\
C_0(1) & =\{j,\omega[i]j,...,{}^j\omega,...,{}^j\epsilon_0,[\omega_1]|j\in\Bbb{N},i=\{1,2,3\}\},\eta=0\\
\Psi_0(1) & =\sup_{j<\omega}(\{{}^j\epsilon_0\})=\epsilon_1
\end{align}
Denote $\zeta_0 = \dfrac{c}{2\sqrt{km}}, \omega = \sqrt{\dfrac km}, x_1= x, x_2=x'$. Transform the equation $x'' + \dfrac cm x' + \dfrac km x =0$ to a linear system of the state vector $\xi = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$
$$ \dot \xi = \begin{bmatrix} 0 & 1 \\ -\omega ^2 & -2\zet...
Over on this post, it was discussed whether or not we should have writing challenges on Meta. The result: we give it a try, and see how it works. That's the point of this post!
Here's how this will work:
The plan was for there to be a (optional, I think) prompt, to help inspire you to write. ...
Show that the power series $\sum_{n=1}^{\infty}\frac{x}{n(1+nx^2)}$ converge uniformly on $\mathbb{R}$.
How should i approach this question? Weirestrass M-test does not work for $|x|>1$ here.
