I have just approached the following series  $$\sum_{n=1}^\infty \sin [\pi(\sqrt5+2)^n]$$ And I already have a question. The $\lim_{n \to \infty}\pi(\sqrt5+2)^n=+\infty$. And the $\lim_{n\to \infty} \sin[\pi(\sqrt5+2)^n]$ does not exist.  Can $\sum a_n$ converge if $\lim_{n\to \infty}a_n$ does not exist? I guess the answer is yes because I know that $\sum a_n$ cannot converge if $\lim_{n\to \infty}a_n\neq 0$. Am I right?    The second part of my question concerns the strategy to use to test convergence of such a series. I can test convergence of series with standard tests. But it seems I need to develop a more sophisticated approach in order to successfully deal with series like this. For example, here, should I use some sort of expansion? If yes, what kind of expansion?  Thank you for your help.
Per the title, does $\displaystyle\sum_{n=1}^{\infty} \sin(\pi(2+\sqrt{3})^n)$ converge? Converge absolutely?   I'm stuck on this question, not sure how to approach it.  Thank you!
Quotient of a local regular ring How can I prove this:     Let $A$ be a local regular ring with maximal ideal $\mathfrak m$ and $x \in \mathfrak m-\mathfrak m^2$. Then $A/(x)$ is a regular ring. Prove also that if $x\in\mathfrak m^2$, $x\ne 0$, the result does not hold anymore.   I don't know how to begin! Thanks to everyone who helps!  But if $d=\infty$? When has Wonder Woman died? I know Wonder Woman has died in canon before, but in which issue did this happen? Show a Form without stealing focus? I'm using a Form to show notifications (it appears at the bottom right of the screen), but when I show this form it steals the focus from the main Form. Is there a way to show this "notification" form without stealing focus? How to convert a std::string to const char* or char* How can I convert an std::string to a char* or a const char*? The question is as below:      Show that each vertex $v$ of a tree $T$ appears $deg(v)-1$ times in the Prüfer encoding of $T$, where $deg(v)$ is the degree of $v$ in $T$.   For this, I think there are 3 scenarios a leaf should be considered for: (1) when it is a leaf as part of a tree; (2) a leaf that has been discarded; (3) the remaining edge. I feel like this will result in a short proof which is only two (or so) statements long, but at the same time, feel it is too simple.   EDIT: I've had a look at this again and narrowed it down to 2 cases:   Case 1: The vertex $v$ in question has $deg(v)\ge2$. $v$ then loses its neighbours as the algorithm is applied repeatedly until $v$ becomes a leaf itseelf. At this point, $v$ is attached to another neighbour, leaving an edge. This means the algorithm can no longer be applied and we have obtained the final Prüfer Encoding for the tree $T$.  Case 2: The vertex $v$ has $deg(v)\ge 2$. 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Here is the code:  \documentclass[12pt]{article}  \usepackage[a4paper, margin = 20mm, top = 30mm, bottom = 25mm]{geometry} \usepackage{xcolor, soul} \usepackage{listings}  \definecolor{codebackground}{rgb}{0.95, 0.95, 0.92}  \lstset {     frame            = tb, % hrule above and below     keepspaces       = true,     columns          = flexible,     basicstyle       = {\normalsize\ttfamily},     escapeinside     = {(*@}{@*)}, % for escaping     backgroundcolor  = \color{codebackground},     showstringspaces = false,     language         = C,     keywordstyle     = \color{blue},     stringstyle      = \color{red},     commentstyle     = \color{teal},     numbers          = left, % {none, left, right}     firstnumber      = 1,     numberstyle      = \scriptsize\color{black},     numbersep        = 5pt }  \let\origthelstnumber\thelstnumber \makeatletter % for stopping line numbers \newcommand*\stopnumber {     \lst@AddToHook{OnNewLine}     {         \let\thelstnumber\relax         \advance\c@lstnumber-\@ne\relax     } } % for starting line numbers \newcommand*\startnumber[1] {     \setcounter{lstnumber}{\numexpr#1-1\relax}     \lst@AddToHook{OnNewLine}     {         \let\thelstnumber\origthelstnumber         \refstepcounter{lstnumber}     } } \makeatother  \begin{document} \begin{lstlisting}[firstnumber = 456] int third_function(const int c) {     /* error checking */     if(c &lt; 0)     {         return;     } (*@ \stopnumber @*)      ...     (*@ \startnumber{512} @*)     /* other stuff */     int d = 4;     char* word = "third function"; (*@ \stopnumber @*)      ...     (*@ \startnumber{678} @*)     return (c * d); } \end{lstlisting} \end{document} Windows 8 hibernation option missing from power options Windows 8 Professional 32-bit running on Laptop. And the Hibernation option is missing from the Power Options in the System Settings. Even after I enable hidden power features by giving admin privileges.  How can I fix this?    Also the option to enable hibernation is missing from the advanced power plan options.      SOLUTION UPDATE:  Thanks to  I have found that you need to enable hibernation using a command prompt.   Get to the Start menu, type 'CMD' this will begin the search. Press the CTRL+SHIFT+ENTER to start the command prompt with administrative privileges. Type powercfg -h on and press enter How to display all the tables with more information (create date, size,...) in a MySQL database? 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