When I saw $$\int_0^\infty\frac{|\sin t|}{t}\;dt$$ integral I remembered Laplace transform and complex analysis but I couldn't find anything. Is this integral bounded (convergent) and if it is what is it's value? I cannot do anything.
Improper integral $\sin(x)/x $ converges absolutely, conditionaly or diverges?  We have that $$\int_1^{\infty}\frac{\sin x}{x}dx$$ $$u=\frac{1}{x}$$ $$du=-\frac{1}{x^2}dx$$ $$dv=\sin xdx$$ $$v=-\cos x$$ $$\int_1^{\infty}\frac{\sin x}{x}dx=-\cos(1)-\int_1^{\infty}\frac{\cos x}{x^2}dx$$ $$\int_1^{\infty}\frac{\cos x}{x^2}dx&lt;\int_1^{\infty}\frac{1}{x^2}dx$$ So it converges. Now I need to find out if $\int_1^{\infty}|\frac{\sin x}{x}|dx$ converges or diverges.
How to compute $$\sqrt{(1\sqrt{(2\sqrt{(3\dots)})})}$$ &amp; $$\sqrt{(1+\sqrt{(2+\sqrt{(3+\cdots)})})}$$?  I understand that  $$\sqrt{(1\sqrt{(2\sqrt{(3\dots)})})}=(1^{1/2})(2^{1/4})(3^{1/8})\cdots$$  and  $$\sqrt{(1+\sqrt{(2+\sqrt{(3+\cdots)})})}=(1+(2+(3+(\cdots))^{1/8})^{1/4})^{1/2} $$  How to Proceed further? I have a requirement for a custom app I am building to open an attachment on the parent record in a new window on create of a child record.  I have parent object Quality_Assessment__c with child object Chart_Review__c. Quality_Assessment__c has a PDF attached to it in the Notes &amp; Attachments section. When a user clicks to create a new Chart_Review__c record I need to have the PDF attachment on the Quality_Assessment__c record open in a new tab as well as the record create page for the Chart_Review__c record. Looking for help in generating the needed javascript for a custom button to accomplish this.  The 'Quality_Assessment__c' record will have a PDF file attached to it prior to the user clicking the 'New Chart Review' button.  My use case is to have users open the 'Quality_Assessment__c' record and perform what we call a Chart Review. To perform the Chart Review the user would click on a custom button called 'New Chart Review' which would do two things.   create a new 'Chart_Review__c' record displaying the new record edit page  open the PDF attachment from the related 'Quality_Assessment__c' record.    Preferably on click of the button the new record page would open in existing tab and a new tab with the PDF would open simultaneously. How to calculate the division and remainder of a big number manually Let $x = 2017^{2017}$. How can I manually determine:   The remainder of the division of $x$ by $31$ The least significant digit of $x$ I am trying to make a floor plan for my office, have managed to make all of the rectangular components (e.g. the rooms) however some of the desk layouts are circular and I haven't been able to work out how to make exact and measured circles. They need to have a radius of 1.45m. An idempotent operator is compact if and only if it is of finite rank I am travelling to Calgary from Delhi. I have an Air India flight from Delhi to Dubai then British Airways flight from Dubai to London and Air Canada flight from London to Calgary. My query is "Do I require a transit visa." Plz help as I am travelling in four days I had a look at Knuth's The Art of Computer Programming, book 1. In chapter 1, section 1.2.2, exercise 25, he presents the following algorithm for calculating logarithm:  given $x\in[1,2)$, do the following:  L1: [Initialize] Set $y\leftarrow0$, $z\leftarrow x/2$, $k\leftarrow1$.  L2: [Test for end] If $x=1$, stop.  L3: [Compare] If $x-z&lt;1$, go to L5.  L4: [Reduce values] Set $x\leftarrow x-z$, $z\leftarrow x/2^k$, $y\leftarrow y+\log_b(2^k/(2^k-1))$, and go to L2.  L5: [Shift] Set $z\leftarrow z/2$, $k\leftarrow k+1$, and go to L2.  (The algorithm needs an auxiliary table which stores $\log_b2$, $\log_b(4/3)$, and so on, to as many values as the precision of the computer.  Then Knuth concludes: this exercise is to explain why the above algorithm will terminate and why it computes an approximation of $y=\log_b x$.  Well, I see that it works, but I am not able to explain why... Let a be a positive number. Then $\lim_{n \to \infty}[\frac{1}{a+n}+\frac{1}{2a+n}+\cdots +\frac{1}{na+n}]$ I would like a proof of the following result, given on .  For all square matrices $\mathbf{A}$ and column vectors $\mathbf{u}$ and $\mathbf{v}$ over some field $\mathbb{F}$, $$ \det(\mathbf{A}+\mathbf{uv}^\mathrm{T}) = \det(\mathbf{A}) + \mathbf{v}^\mathrm{T}\mathrm{adj}(\mathbf{A})\mathbf{u}, $$ where $\mathrm{adj}(\mathbf{A})$ is the adjugate matrix of $\mathbf{A}$.  Note that $\mathbf{A}$ may be singular. However, the proof given on wikipedia requires that $\mathbf{A}$ is nonsingular. Export Windows 7 search results to a text file How to use an optimization solver to get t-stats and p-values for the estimates? Windows 10 non-persistent installation (live ISO) We've got a Windows 10 installation running in an environment where it can occasionally lose power and restart. However, this sometimes leads to corruption of files.  I've had a look at Windows "To-Go" but this seems to be persistent. And I've also checked out virtualisation, but that would require us to run a non-persistent Windows 10, in another Windows 10 environment.  Is it possible to install Windows 10 as a "non-persistent live CD"? So that a user will have a seemingly fresh Windows 10, but doesn't require to re-install it at each startup? How can I give other drives and partitions short, meaningful names in nautilus? In nautilus and Unity the external drive has a nice short descriptive name "mcat", but all partitions on the 4 internal drives are displayed as a size (834 GB filesystem) or a huge 32-character string form of a GUID:  I'm guessing the external drive is nice, short, sweet and readable because that drive may have no partitions (well, just one I guess) and that name may be the drive label, whereas partitions usually don't have names. That may explain my problem, but doesn't solve it.  Is there some way to give reasonable names to these partitions in nautilus and Unity? Is it ever acceptable to have a  in your C or C++ application?  What if you allocate some memory and use it until the very last line of code in your application (for example, a global object's destructor)? As long as the memory consumption doesn't grow over time, is it OK to trust the OS to free your memory for you when your application terminates (on Windows, Mac, and Linux)? Would you even consider this a real memory leak if the memory was being used continuously until it was freed by the OS.  What if a third party library forced this situation on you? Would refuse to use that third party library no matter how great it otherwise might be?  I only see one practical disadvantage, and that is that these benign leaks will show up with memory leak detection tools as false positives. Allow moderators to see delete/undelete activity on users' profiles We get auto flags for 'possible vandalism' related to deletes, and these can be helpful at times in finding 'rage quit' users, or other potential problems.  But often, I would like to know if there is a prior pattern with the user doing this. While we can see posts of theirs which have been flagged, and perhaps find previous auto flags this way, it would be nice to see an overall view of their self-deletions and undeletions; mostly just to identify patterns.  Such a feature would be nice if it was activity centered, rather than post-centered. That is; each delete/undelete action should be shown. So we could easily see, for example, if someone went back-and-forth on a single post.  The "activity" tab in the profile would be a logical place to show these events.  I think this would be best to be limited to instant self-deletions, not deletion votes. I want to run a daily incremental backup and append the day of the week to the file name so I end up with:  backup_mon.bak backup_tue.bak  etc.  Not really bothered about what exactly is appended - 0 - 6 is fine.  How do I go about this? The Wiki page on  says     The pair $(m, m+2)$ is twin prime, iff $4((m-1)! + 1) \equiv -m \pmod {m(m+2)}$.   This is obviously connected to Wilson's Theorem. Can anybody provide a proof for that, along the lines of thought given here, ? System log full of session and slice messages Numerical phenomenon. Who can explain? I was doing some software engineering and wanted to have a thread do something in the background to basically just waste CPU time for a certain test.  While I could have done something really boring like for(i &lt; 10000000) { j = 2 * i }, I ended up having the program start with $1$, and then for a million steps choose a random real number $r$ in the interval $[0,R]$ (uniformly distributed) and multiply the result by $r$ at each step.   When $R = 2$, it converged to $0$. When $R = 3$, it exploded to infinity.    So of course, the question anyone with a modicum of curiosity would ask: for what $R$ do we have the transition. And then, I tried the first number between $2$ and $3$ that we would all think of, Euler's number $e$, and sure enough, this conjecture was right. Would love to see a proof of this.    Now when I should be working, I'm instead wondering about the behavior of this script.   Ironically, rather than wasting my CPUs time, I'm wasting my own time. But it's a beautiful phenomenon. I don't regret it. $\ddot\smile$ Let $E=l^p=\left\{x=\left(x_n\right): x_n\in\mathbb{R}, \sum_{1}^{\infty}\left|x_n\right|^p&lt;\infty\right\}$ with $1\le p&lt;\infty$, $\left\|x\right\|_E=\left(\sum_{1}^{\infty}\left|x_n\right|^p\right)^{1/p}.$ Let $\left(\lambda_n\right)$ is a bounded sequence in $\mathbb{R}$ and consider the operator $T$ defined by $$T(x)=\left(\lambda_1x_x,\ldots,\lambda_nx_n,\ldots\right)$$ where $x\in l^p$. Prove that $T$ is a compact operator from $E$ to $E$ iff $\lambda_n\rightarrow 0$.  Can anyone give me some hints to solve this problem? Thank you!