I have come across the following problem in a calculus textbook:  We are asked to find the values of $ \beta \in \mathbb{R} $ for which the improper integral $$ \int_{1}^{\infty}x^2 \cos({x^\beta})dx $$  converges. In fact, I have no idea at all where to start on this it doesn't look like something familiar at all. All help is appreciated.
I need to show that $$ \int\limits_1^{\infty}x^\alpha\sin(x^\beta)dx $$ converges if and only if $\alpha + 1 &lt; \beta$. I used substitution and integration to get $$ \lim\limits_{t\to\infty}\int\limits_1^tx^\alpha\sin(x^\beta)dx \overset{u = x^\beta}{\underset{du = \beta x^{\beta-1}}{=} }\lim\limits_{t\to\infty}\left(\frac{1}{\beta}\int\limits_1^{t^\beta}u^{\frac{\alpha + 1 - \beta}{\beta}}\sin{u}du\right) $$ If $\alpha + 1 -\beta &lt; 0$, then we have a product if a function with bounded anti-derivative and a differentiable decreasing function which tend to 0, and from Dirichlet's criterion, the integral converges.   For the converse, I used integration by parts $$  = \frac{1}{\beta}\lim\limits_{t\to\infty}\left(\left.-u^{\frac{\alpha + 1-\beta}{\beta}}\cos{u}\right|_1^{t^\beta} + \frac{\alpha+1-\beta}{\beta}\int\limits_1^{t^\beta}u^{\frac{\alpha + 1-2\beta}{\beta}}\cos{u}du\right) $$ For $\alpha + 1 -\beta\geq 0$, I get that the first term indeed diverges, but this is not enough to show total divergence. How can one show it?
I have an excel file with 2 attribute columns, 1-feature info 2-Number of color (RGB) that the current feature should be symbolize. This file file can be joined with the layer using the "feature info" field which is unique (join by attribute table). So i have as attribute the color number attribute for every polygon feature.  How can  i use this this information to symbolize every feature with their attributed color already imported from the excel sheet? Why is it important to override GetHashCode when Equals method is overridden? I am seeing quite a few conversions which have as source "accounts.google.com" (almost 40%).  AFAIK, in the new Universal Analytics as soon as a user has a Referrer set a new session is started. So if a user is getting on my page, and creates an account using social auth (e.g. Google accounts) when being redirected back, will this create a new session? I guess this would explain that there are many conversions coming from accounts.google.com - is it safe to enter accounts.google.com into the Referral Exclusion List? And more - what about Facebook logins, those would show up as facebook.com, but i probably don't want to exclude facebook.com as referrer?  Any ideas how to deal with UA referrers and social logins? Our company would like to understand the effects:   If we have single table with Primary Key Identity column, int primary key IDENTITY(1,1),  and the Application is conducting multi-threading parallel processing, trying to conducts 1000s of simultaneous inserts on one table.   Would the identity primary key column fail, with a primary key violation in any way, for example: processors are trying to input the same identity number?  Or was SQL designed to be successful even in multithreading parallel insert?  Note: IDENTITY_INSERT is OFF What determines who gets or becomes an MVP? This hasn't been asked yet, but what determines how or who gets an MVP award in CS:GO? After a match, there's always a little blurb that first states who got MVP first, and then a random fact (Player X got Y knife kills, Player X lasted Y rounds without dying, etc.).  Do these two factors both "factor" into getting an MVP award? Is it a set percentage of enemies you kill? Is there a priority list (ie, planting bomb, defusing, etc.)? Imagine this directory structure:  app/    __init__.py    sub1/       __init__.py       mod1.py    sub2/       __init__.py       mod2.py   I'm coding mod1, and I need to import something from mod2. How should I do it?    I tried from ..sub2 import mod2 but I'm getting an "Attempted relative import in non-package".  I googled around but found only "sys.path manipulation" hacks. Isn't there a clean way?      Edit: all my __init__.py's are currently empty    Edit2: I'm trying to do this because sub2 contains classes that are shared across sub packages (sub1, subX, etc.).  Edit3: The behaviour I'm looking for is the same as described in  (thanks John B) How do you show a hidden folder (whose name starts with .) in File Manager a.k.a. Nautilus on Ubuntu?  I am somewhat new to Ubuntu and can't seem to find the "view" menu. I'm building an HTML UI with some text elements, such as tab names, which look bad when selected. Unfortunately, it's very easy for a user to double-click a tab name, which selects it by default in many browsers.  I might be able to solve this with a JavaScript trick (I'd like to see those answers, too) -- but I'm really hoping there's something in CSS/HTML directly that works across all browsers. Prove that the power set of an $n$-element set contains $2^n$ elements Theorem. Let $X$ denote an arbitrary set such that $|X|=n$. Then $|\mathcal P(X)|=2^n$.      Proof. The proof is by induction on the numbers of elements of $X$.      For the base case, suppose $|X|=0$. Clearly, $X=\emptyset$. But the empty set is the only subset of itself, so $|\mathcal P(X)|=1=2^0$.      Now the induction step. Suppose $|X|=n$; by the induction hypotesis, we know that $|\mathcal P(X)|=2^n$. Let $Y$ be a set with $n+1$ elements, namely $Y=X\cup\{a\}$. There are two kinds of subsets of $Y$: those that include $a$ and those that don't. The first are exactly the subsets of $X$, and there are $2^n$ of them. The latter are sets of the form $Z\cup\{a\}$, where $Z\in\mathcal P(X)$; since there are $2^n$ possible choices for $Z$, there must be exactly $2^n$ subsets of $Y$ of which $a$ is an element. Therefore $|\mathcal P(Y)|=2^n+2^n=2^{n+1}$. $\square$      From the above explanation, I don't understand why the set that contains $\{a\}$ will contain $2^{|n|}$ elements when it should clearly be $2^{|1|}$.  The construction of a new set $S$ is the union of the old set with cardinality $n$ and an new element $\{a\}$, therefore the set that do not contain $\{a\}$ still has cardinality $n$ and the set that contains $\{a\}$ is just $\{a\}$, one element...  Can someone please elucidate? Which Stack Exchange site is best for AutoCAD questions? I just synced my Mountain Lion Mail app with my Exchange work email account in order to have it locally on my machine (rather than continuously going through the web interface). However, now Mail fetches all the mails since the very beginning (which are a couple of years).  Is there a way (like on iOS devices) to tell Mail to only sync the last month or something configurable amount based on message count, download size or other similar restriction to prevent all messages from being stored locally? Is a sentence always grammatically incorrect if it has no verb? Is the following grammatically correct? My friend says the second sentence is grammatically incorrect, but couldn't explain why.     I have always been fascinated by statistics. The different ways in   which you could look at data and infer knowledge from it. (Ubuntu 13.04 with GNOME Shell, upgraded, GNOME version 3.8) Recently, I upgraded Ubuntu to 13.04. the default desktop is gnome-shell. Except Nautilus, other software border doesn't disappear when I make them to be maximized. I checked the problem for these software:  LibreOffice Firefox Rhythmbox thunderbird GNOME-Terminal  On the other hand there are three bars on maximized mode:  GNOME Shell panel  Title-bar for software which contains - + x (minimise, maximise, close) buttons.  Software menu-bar  Number &quot;2&quot; is extra. Is a package missing? How to run a command that involves redirecting or piping with sudo? Workflow email with link to document in library Mars needs water and women I read this book in high school too many years ago for me to think about. Martian colonist are raiding earth's oceans for water and because of the conditions on Mars where only male children are conceived, they are also on the look out for young women as potential mates. Our main character while on a raid, bags not only a young woman but her brother as well. Returning to Mars they find that Earth has attacked Mars can't remember much more. A Linear Diophantine Equation is of the following form:                                                                                        $Ax+By+C=0$, where $x_1 \leq x \leq x_2$ and $y_1 \leq y \leq y_2$.  If the value of $A,B,C,x_1,x_2,y_1,y_2$ are given and $x_1 \leq x_2$ and $y_1 \leq y_2$, then how many solutions can be found? How can I find out the total number of solutions according to the above condition. Best way to list files in Java, sorted by Date Modified? I want to get a list of files in a directory, but I want to sort it such that the oldest files are first.  My solution was to call File.listFiles and just resort the list based on File.lastModified, but I was wondering if there was a better way.  Edit: My current solution, as suggested, is to use an anonymous Comparator:  File[] files = directory.listFiles();  Arrays.sort(files, new Comparator&lt;File&gt;(){     public int compare(File f1, File f2)     {         return Long.valueOf(f1.lastModified()).compareTo(f2.lastModified());     } }); complex number question involving modulus Let $z, w$ be distinct complex numbers. Show that if $|z| = 1$ or $|w| = 1$, then $$\frac{|z − w|}{|1 − z^*w|} = 1$$ [Hint: Note that $|a|^2 = aa^*$.]  Hey guys, couldn't get my thinking cap on for this question. Any helpful input? Will appreciate it! I want to produce four groups (high/high, high/low, low/high and low/low) using two continues variables and compare these groups in terms of a few dependent variables. I know that cluster analysis (k-means and hierarchical) is a type of 'end' analysis which is the main goal of the paper to reach such clusters and then discuss the nature of the clusters. However, I am willing to use this technique (k-means by using k=4) as a kind of preliminary analysis to produce the groups. I know that scatter plot and means splits are other alternatives but they have their own disadvantages that cluster analysis may resolve. What do you think?