Intereting Posts

Find a limit in an efficent way
Is there any palindromic power of $2$?
How far are the $p$-adic numbers from being algebraically closed?
Taylor series for different points… how do they look?
Cauchy-Riemann equations in polar form.
Have I found an example of norm-Euclidean failure in $\mathbb Z $?
Products of CW-complexes
Help with $\lim_{x, y\to(0, 0)} \frac{x^2y}{x^4+y^2}?$
How does one denote the set of all positive real numbers?
How to use LU decomposition to solve Ax = b
Probability in multiple choice exams
Calculate an integral in a measurable space
Explicit solution of the recursion $x_n = x_{n-1}^2 – 2$ with $x_0>2$
Probability density function of $X^2$ when $X$ has $N(0,1)$ distribution
How much Math do you REALLY do in your job?

So I recently asked a question about convergence of $\sum_n |\sin n|^{cn}$ for arbitrary $c > 0$ and it turns out that the terms of the series don’t even converge, for any $c > 0$, so the series is always divergent. But what about $\sum_n |\sin n|^{cn^2}$ for $c > 0$? Are there $c$ so that the series converges, and if there are $c > 0$ such that the series diverges, do the terms of the series still converge?

If that’s too hard, what if we replace $n^2$ in the exponent by $n^\alpha$ for some different $\alpha > 1$? Which $\alpha$ do we know the answer for?

- Selection of $b_n$ in Limit Comparison Test for checking convergence of a series
- Does convergence in distribution implies convergence of expectation?
- What is the radius of convergence of $\sum z^{n!}$?
- What are conditions under which convergence in quadratic mean implies convergence in almost sure sense?
- Is the space $C$ complete?
- Is this:$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$ a convergent series?

- $\lim_{n \to \infty} \mid a_n + 3(\frac{n-2}{n})^n \mid^{\frac1n} = \frac35$. Then find $\lim_{n \to \infty} a_n$.
- weak convergence of probability measures and unbounded functions with bounded expectation
- Evaluate $\int_0^{{\pi}/{2}} \log(1+\cos x)\, dx$
- Cauchy nets in a metric space
- Proving the series $\sum_{n=0}^{\infty}\sin(e\pi n!)$ converges
- How discontinuous can a derivative be?
- What is the cardinality of the set of all non-measurable sets in $\Bbb R^n$?
- Need Suggestions for beginner who is in transition period from computational calculus to rigorous proofy Analysis
- Is $ \sin: \mathbb{N} \to \mathbb{R}$ injective?
- Proving uniform continuity on an interval

- Countable product of Polish spaces
- Zeros of Fourier transform of a function in $C$
- How can I interpret “energy” in signals?
- Canonical divisor on algebraic curve
- Show that every group $G$ of order $175$ is abelian and list all isomorphism types of these groups
- Prove the $n$th Fibonacci number is the integer closest to $\frac{1}{\sqrt{5}}\left( \frac{1 + \sqrt{5}}{2} \right)^n$
- Help me put these enormous numbers in order: googol, googol-plex-bang, googol-stack and so on
- AGM Inequality Proof
- How to find $n$'th term of the sequence $3, 7, 12, 18, 25, \ldots$?
- How is it possible for two random variables to have same distribution function but not same probability for every event?
- The kernel and range of the powers of a self-adjoint operator
- Calculate in closed form $\int_0^1 \int_0^1 \frac{dx\,dy}{1-xy(1-x)(1-y)}$
- Estimate variance given a sample of size one (7th Kolmogorov Student Olympiad)
- Product rule intuition
- Summing divergent series based on physics