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Write it as $a_n(\alpha)$ to emphasize the dependence on $\alpha$. For any $\epsilon > 0$, $U(n,\epsilon) = \{\alpha: |a_n(\alpha)| < \epsilon\}$ is an open set containing $k/m$ for any integers $k,m$ …

For example, it is true if $C$ contains a representative of every equivalence class of bijections of $\mathbb N$, where $f \sim g$ if there exist $N,M$ such that $|f(i) - g(i)| < M$ for all $i > N$.

The answer is no. For any $T \subseteq \mathbb N$, let $$ A(T) = \sum_{n \in T} a_n (-1)^{B_n}$$ Now $A = A(T) + A(\mathbb N \backslash T)$ where $A(T)$ and $A(\mathbb N \backslash T)$ are independen …

Non-tangential convergence. This is a common theme in e.g. boundary values of harmonic functions, Hardy-Littlewood maximal functions, etc.

A solution of $a = 1/2^a$ is transcendental. First note that $a$ is not an integer. It can't be a non-integer rational, because $2^a$ for positive rational $a$ is an algebraic integer, and the only …

No, in fact in the Banach space $c_0$, for any sequence $\epsilon_n$ of positive numbers decreasing to $0$, we can choose a normalized weakly-null sequence $x_n$ such that for every nonzero bounded li …

Your differential equations do have closed-form solutions. $f(t)$ is a rather complicated expression involving Bessel functions, while $$ g \left( t \right) ={\frac {{{\rm e}^{- \left( \gamma-\beta …

Any function that is in $L^1$ but not in $L^\infty$ will have some point in whose neighbourhood it is unbounded, and the Fourier series is likely to diverge at such a point. A simple example is $$ f( …

Yes, using Runge's theorem. I'll assume wlog your $D$ is the unit disk $\{z: |z| \le 1\}$. Given positive integer $n$, take $$ K_n = \{z: (|z| \le 1 \ \text{or}\ 1 + 1/n \le |z| \le n)\ \text{and} …

If $x > 0$, let $r = 1/|1 + 2 i/x| = 1/\sqrt{1 + 4/x^2}$ and the $n$'th term on the right of the second sum is dominated by $2/((2n-1)r^{2n-1})$. On the other hand, for large $n$ the $n$'th term in t …

$T_n \sim N(\mu, \sigma/\sqrt{n})$, and $e^{T_n}$ has a lognormal distribution. Its mean and variance are $$\eqalign{\exp(\mu &+ \sigma^2/(2n))\cr &= \exp(\mu) \left(1 + \dfrac{\sigma^2}{2n} + O\left( …

If, as Willie Wong suggests, the question is about the lim inf of $b^n (n a - \lfloor n a \rfloor)$, that is almost always $+\infty$ but generically $0$ (i.e. there is a dense $G_\delta$ set of $a$'s …

Consider the Jordan canonical form $Q = U J U^{-1}$ for $Q$. Thus $w^T Q w = \sum_B w_B^T B v_B$ for Jordan blocks $B$, where $w_B$ and $v_B$ are the entries of $U^T w$ and $U^{-1} v$ corresponding …

If $f(x,y) = \ln(1+x) + \ln(1+y)$, and $p = -1 - 2 W_{-1}(-1/(2 \sqrt{e}))$, it is easy to verify that $f(p,p) = p$. Moreover, it appears numerically that $\|(x_3,x_4) - (p,p)\| < \|(x_1, x_2) - (p,p …