I was recently asked to find  $$\liminf_{n-&gt;\infty} n|\sin(n)|.  n \in \mathbb{N}$$  I suspect that the answer is $0$, but I'm not quite sure how to prove it. I know that $\{ \sin(n) : n \in \mathbb{N}\}$ is dense on the real line, but from this fact is not easy to me to deduce that $\{ n\sin(n) : n \in \mathbb{N}\}$ is also dense. Maybe one can use this inequality $$n |\sin n| \le |n^2-nk\pi|$$  for every $n,k \in \mathbb{N}$ and prove that $|n^2-nk\pi|$ can be as small as desired, but that seems hard to me.
10261
How would you evaluate the limit inferior of the sequence $n\,|\mathopen{}\sin n|$? That is, $$\liminf\limits_{n\to\infty} \ n \,|\mathopen{}\sin n|$$    Edit. Let $\mu$ be the irrationality measure of $\pi$. Since $\mu$ is not known, I will split the question:   Assuming $\mu &gt; 2$, what is $\liminf\limits_{n\to\infty} \ n\,|\mathopen{}\sin n|$? Assuming $\mu = 2$, what is $\liminf\limits_{n\to\infty} \ n\,|\mathopen{}\sin n|$?     I kind of like its graph...
81400
$$\sum_{n=1}^\infty \sin^{[n]}(1)$$ Where by $\sin^{[n]}(1)$ we mean $ \sin\left(\sin\left(\dots\sin(1)\right)\right)$ composed $n$ times.   Have tried the divergence test, which fails.  Have tried Ratio test, also fails, as the limit is 1.  Integral test, or root test do not seem promising. Help is appreciated
19573
Let $a &gt; 0$ and let $k \in \mathbb{N}$. Evaluate: $$\lim_{n\to \infty}a^{-nk}\prod ^k_{j=1}\left(a+\frac{j}{n}\right)^n$$  Clueless on this problem. Seek your help.
4575
I'm trying to prove  the following claim:     If $f$ is continuous  and $\displaystyle\int_a^\infty |f(x)|\;dx$ is finite then $\lim\limits_{ x \to \infty } f(x)=0$.   Here the counter example of all these sine functions won't do, since  their infinite integral does not converge absolutely.   Any hints for a proof?
95017
I'm very confident that  $$\sum_{p \ \text{prime}} \sin p $$  diverges. Of course, it suffices to show that there are arbitrarily large primes which are not in the set $\bigcup_{n \geq 1} (\pi n - \epsilon, \pi n + \epsilon)$ for sufficiently small $\epsilon$. More strongly, it seems that $\sin p$ for prime $p$ is dense in $[-1,1]$.   This problem doesn't seem that hard though. Here's something that (to me) seems harder.      If $p_n$ is the nth prime, what is $$\limsup_{n \to +\infty} \sum_{p \ \text{prime} \leq p_n} \sin p?$$   What is $$\sup_{n \in \mathbb{N}} \sum_{p \ \text{prime} \leq p_n} \sin p? $$   Of course, we can ask analogous questions for $\inf$.   I'm happy with partial answers or ideas. For example, merely an upper bound.
49258
The limit is   $$\lim_{x\rightarrow0} \frac{x-\sin_n(x)}{x^3},$$  where $\sin_n(x)$ is the $\sin(x)$ function composed with itself $n$ times:  $$\sin_n(x) = \sin(\sin(\dots \sin(x)))$$  For $n=1$ the limit is $\frac{1}{6}$, $n=2$, the limit is $\frac{1}{3}$ and so on.  Can we define a recurrent relation upon that given hypothesis? Also, how do I involve $n$ into calculation, because the final limit will depend on it?  Any suggestions on how to tackle this?  Thank you!
28951
I need to prove that $$\sum_{n=1}^{\infty} {\sin{100n}} \; \text{diverges}$$ I think the best way to do it is to show that $\lim_{n\to \infty}{\sin{100n}}\not=0$. But how do I prove it?
123466
I have to show that for every $x \in \mathbb{R}$ and every $n \in \mathbb{N}$ $$|\sin(nx)| \le n|\sin(x)|  $$ In the previous exercise, I have showed that $|\sin(x)|≤|x|$ with the use of the mean value theorem. I think that I cannot use this approach this time. I also tried to write $\sin(nx)$ as a series expansion but that doesn't work either.     Does anyone know how I can solve this?
99236
How to prove $\sum_{n=1}^{\infty}\frac{|\sin{(n\theta_0)}|}{n}$ diverges for given $\theta_0\in (0,\pi)$?   I would appreciate any help.
103599
can anyone help me with this, I got stuck on it.  let $a_n$ and $b_n$ (for $n$ a non-negative integer) be two sequences of real numbers such that $\lim_{n\to \infty}a_n=a$ and $\lim_{n\to \infty}b_n=b$. prove that $\lim_{n\to \infty} \frac{a_0b_n+a_1b_{n-1}+....+a_nb_0}{n}=ab$.
101258
How to prove that $$\lim_{n\to\infty}{\sin{100n}}$$ doesn't exist?    Some possible approaches:   It would be enough to find two subsequences $n_{k}$ that converge to two different numbers. But it's not clear how to find $n_k$ so that $\sin 100n_k$ converge.  Show that $\sin (100(n+1))-\sin 100n$ does not approach $0$. This is not obvious, either.
56323
Let $f,g\in PC(2\pi)$ continuous. Show that $$\lim_{n\to \infty}\int_{-\pi}^{\pi}f(x)g(nx)\,dx=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)\,dx\int_{-\pi}^{\pi}g(x)\,dx$$ Where if $f\in PC(2\pi)$ then $\int_{-\pi}^{\pi}f(x)\,dx=\int_c^{c+2\pi}f(x)\,dx \quad\forall c\in\mathbb{R}.$ I've been struggling with this proof. Any suggestions would be great!
98906
Show that $\lim_{n\to \infty}\sin{n\pi x} =0$ if $x\in \mathbb{Z},$ but the limit fails to exist if  $x\notin \mathbb{Z}.$ 1st part If $x\in \mathbb{Z}$ then $\sin{n\pi x}=0$ for all $n,$ giving the first part. Edit:  2nd part If $x\notin \mathbb{Z},$ I want to show that the limit doesn't exist. How to do that?
15989
This is half of Theorem 3.19 from Baby Rudin. Rudin claims the proof is trivial. What I've come up with so far doesn't seem trivial, however, and is probably also wrong (my problem with it is pointed out below). This makes me wonder whether I'm overlooking some useful fact and/or using an unprofitable approach.  Theorem. If $s_n \leq t_n$ for $n \geq N$, where $N$ is fixed, then  $$ \liminf_{n \rightarrow \infty} s_n \leq \liminf_{n \rightarrow \infty} t_n. $$  Proof. Suppose that $s_n \leq t_n$ if $n \geq N$, but that  $$ \liminf_{n \rightarrow \infty} s_n &gt; \liminf_{n \rightarrow \infty} t_n. $$  Let $E_s$ denote the set of all subsequential limits of $\{s_n\}$, and let $E_t$ denote the set of all subsequential limits of $\{t_n\}$. Then  $$ \inf E_s &gt; \inf E_t. $$  This implies that there exists some $x \in E_t$ such that $\inf E_s &gt; x &gt; \inf E_t$, since otherwise $\inf E_t$ would not be the greatest lower bound of $E_t$. Hence some subsequence of $\{t_n\}$, say $\{t_{n_i}\}$, converges to $x &lt; \inf E_s$.  Lemma. (from Rudin) If $x &lt; \liminf_{n \rightarrow \infty} s_n$, then there exists an integer $N$ such that if $n \geq N$, then $s_n &gt; x$.   By the lemma, there exists an integer $N_0$ such that if $n \geq N_0$, then $s_n &gt; x$.   Now, let $\epsilon = \inf_{n \geq N_0} \{s_n - x\}$. This is where I think my proof breaks down. Can't $\epsilon$ be zero? Then, since $\{t_{n_i}\}$ converges to $x$, there exists an integer $N_1$ such that if $n_i \geq N_1$, then $|t_{n_i} - x| &lt; \epsilon$. But this means that, if $n_i \geq \max\{N, N_0, N_1\}$, we have both  $$ s_{n_i} &gt; t_{n_i}, $$  as well as $s_{n_i} \geq t_{n_i}$, a contradiction.
88366
Here $n$ belongs to natural numbers. Firstly, I proved the relation by putting $n = 1$ . Then, taking $$|\sin(mx)| \leq m|\sin x|$$ true, I had to prove $$|\sin(m + 1)x| \leq (m + 1)|\sin x|$$  Now, here I got stuck. How to prove it?? Please help.
96706
Let $\{x_n \}$ be a sequence of real numbers and let $y_n = \frac{(x_1 + x_2 + ... + x_n)}{n}$.   (a) Prove that $\liminf x_n \le \liminf y_n \le \limsup y_n \le \limsup x_n$  (b) Give an example of a sequence $\{x_n\}$ for which all inequalities of part (a) are strict.  I honestly have no idea where to start on this. I can observe some of the easier things such as $\lim \inf x_n  \le \lim \sup x_n$ Any hints would be appreciated.
11394
Prove that the $\lim_{n\to \infty} r^n = 0$ for $|r|\lt 1$.   I can't think of a sequence to compare this to that'll work.  L'Hopital's rule doesn't apply.  I know there's some simple way of doing this, but it just isn't coming to me. :(