$$\tan\left(\sum_{r=1}^{\infty} \arctan\left(\dfrac{4}{4r^2  +3}\right)\right)= ? $$   I wrote it in the form:   $$\tan\left(\sum_{r=1}^{\infty} \arctan\left(\dfrac{\dfrac43}{\dfrac{4r^2}{3}  +1}\right)\right)$$ and tried to use:  $$\arctan x- \arctan y = \arctan\left(\dfrac{x-y}{1+xy}\right)$$ but that trick doesn't help here. How to go about solving this problem then?
I have to find the limit of this sum: $$\sum_{r=1}^{\infty}\cot ^{-1}(r^2+\frac{3}{4})$$  I tried using sandwich theorem , observing:  $$\cot ^{-1}(r^3)\leq\cot ^{-1}(r^2+\frac{3}{4})\leq\cot ^{-1}(r^2)$$  Now when I was calculating the limit of left hand expression, I convert it to $\tan^{-1}$, by using:  $$\tan^{-1}\frac{1}{x} = \cot^{-1}x$$  but couldn't sum up the terms of arctan series.     How can I proceed?   Is there any better way ?
Proving convergence of sums: If $\sum_{n=1}^{\infty}a^2_n$ and $\sum_{n=1}^{\infty}b^2_n$ converge then $\sum_{n=1}^{\infty}a_nb_n$ converges Does it increase security if the CBC chunks of a AES-256 encrypted bytestream are uploaded to different clouds? I want to backup my data according to the 3-2-1 rule. I am already back up to 2 different drives (2 different systems) at home. In addition, I would like to use cloud storage (Google, Amazon, etc.). Considering the future weakening of cryptography by advancements in computing power (e.g. key guessing by quantum computing) I want my data to be as safe as possible because once it's uploaded it's prone to decryption in the future (i can't guarantee the data is deleted if I delete it from the cloud(s)). Does it increase the security of my data if I split the CBC chunks to let's say 2 clouds? Uploading the even-numbered CBC elements to e.g. Google and the odd-numbered elements to Amazon. Since one storage hoster is constantly missing the next part of the CBC-Chain (1/2 in total) shouldn't this increase security of the data even if the AES key is known by the hoster(s)? How to insert text before the first line of a file? How to install a Brother dcp-135c printer? Recently moved over onto Ubuntu. First time for anything like this for me as I'm coming from Windows but I'm here and ready to learn. The only problems Ive experienced so are saving in Libre office doc as a word doc file and then losing the formatting upon receival via email. However the more pressing issue is getting my old Brother DCP 135 c to work. Ive scanned a few help blogs and have found them frankly quite frightening.  Would it be possible for someone to write u a step by step guide (including explanations of terminology)  For those of us new to this magic world? Does removing an iPhone from my icloud account also erase the data? How does the NSAutoreleasePool autorelease pool work? As I understand it, anything created with an alloc, new, or copy needs to be manually released. For example:  int main(void) {     NSString *string;     string = [[NSString alloc] init];     /* use the string */     [string release]; }   My question, though, is wouldn't this be just as valid?:  int main(void) {     NSAutoreleasePool *pool;     pool = [[NSAutoreleasePool alloc] init];     NSString *string;     string = [[[NSString alloc] init] autorelease];     /* use the string */     [pool drain]; } Chrome address bar no longer shows protocol or www subdomain A couple of days ago I noticed that for all websites, chrome omits from the address bar the protocol and the subdomain if it is www. My co-workers report that they are not seeing this. Chrome version is 67.0.3396.87.  For example, if I type into the address bar https://www.google.com (without the quotes), Chrome immediately displays the chip "Secure" followed by google.com. See attached screenshots.      I have looked in vain for Chrome settings that affect this, plus the chrome://flags page and chrome://net-internals page.  I did not do anything consciously to make this change.  Any ideas?  EDIT: Suggestion from MrWhite that Ominbox UI Hide Stead-State URL Scheme and Trivial Subdomains had become enabled: good suggestion, except that the setting is still at “Default.” See screenshot:    However, as a shot in the dark, I am going to play with the setting to see what happens.  First I set it to “Disabled.” That restored the previous behavior of the address bar, which is to not to hide the protocol and “trivial subdomains.” Then I changed it back to “Default,” and it was hiding them again. One would suspect that somehow the default has been changed, except that co-workers are not reporting this issue.  Oh, well. I’ll just leave it as Disabled for now.  I would be ready to accept the answer from MrWhite, because it provides a workaround. Yet will leave it open for a bit longer, because it does not really answer why this behavior has changed, and why it has to be overridden.  EDIT 2: Chrome version is 67.0.3396.87. Does voltage drop occur with zero load and zero resistance Present or Past tense to describe a past condition which is still present? Scenario: My friend John has a personality disorder, intensely paranoid for example.  Can I say:     There were three events last year that told me John had a problem.   or can I say:     There were threee events last year that told me John has a problem. Accepted answer appearing below top voted On , sorted by votes, the accepted answer is #3 in order of votes.  It appears as number 3 in the list as well. Is this new behavior intended? I am having trouble finding the derivative of the following:  $$y = x^x+x^3+3^x+3^3$$  $$\frac{dy}{dx}= x \times x^{(x-1)}+3x^2+3^x\ln3+0$$  I think the $x^x$ part is wrong. Any help would be appreciated. When I press the power button on my Epson WorkForce WF-3520 printer, it starts up normally, but shuts down within 30 seconds. Ten seconds later, it starts itself and the stop/start cycle is repeated again and again. This has only started happening today (6th December 2016). Fancy Chapter Headings Style I would like make a fancy chapter like the figure.    However, i made this code  \documentclass[12pt,a4paper]{book}   \usepackage{lmodern} \usepackage{titlesec} \usepackage{microtype} \usepackage{tikz}    \titleformat{\chapter}[display]   {\normalfont\bfseries\color{black}}   {%\filleft%     \begin{tikzpicture}     \node[       outer sep=-10pt,       text width=2.5cm,       minimum height=3cm,       fill=black,       font=\color{white}\fontsize{80}{90}\selectfont,       align=center       ] (num) {\thechapter};     \node[       rotate=0,       anchor=south,       font=\color{black}\Large\normalfont       ] at ([yshift=43pt,xshift=30]num.west)  {\textls[180]{\textsc{\chaptertitlename}}};       \end{tikzpicture}%  }    {-90pt}   {\titlerule[2.5pt]\vskip3pt\titlerule\vskip4pt\LARGE\sffamily \raggedright }    \begin{document} \chapter{The Singular Sturm-Liouville Problems in Spectral Parameter}    \end{document}      that produces this one    How can I produce the chapter like the First Figure? When I'm not exploring fresh caves or lands in minecraft, I'm busy trying to find deposits of diamond, gold and iron deep in the earth. So far I've just been digging around just above bedrock on a whim to look for these rare minerals but I've started thinking that I should be taking a more controlled, efficient approach to mining.  What is the optimal mining strategy that yields the most rare minerals while minimizing blocks removed and time spent digging and traveling?  Edit: The existing answers are great for mining, but no one has addressed travel time to and from the mine which is usually ~60 blocks downward and incredibly tedious. Solutions? I know it existed as sort of a folder in Windows Explorer in at least Windows XP, but I just can't seem to find it in Windows 7...  How can I take a picture with my webcam in Windows 7 without using any 3rd party tools? So, I know that there is a race in Lord of the Rings of Peredhil  (half elf). Also, in the Hobbit (film trilogy, not book), there is a romance between a dwarf and an elf.  My question is, how do man/elf or dwarf/elf offspring work? Can they not be born? Are they born infertile because they are from parents of two different species? Are elves, dwarves, and men only different "races" of one single primordial species, and can thus reproduce any number of ways? To my knowledge, this isn't addressed in the books or films. Isolate OSX Dock per Space How can I make it so that each space has its own isolated dock?  Basically what I'm looking for is:   The dock in one space should not show the icon of an app that was open in another space. If I have the same program open in two different spaces, clicking on the icon in one space should not go to the app in the other space.   Is this possible natively, or with a third party program? Fixing scrolling in nano running in tmux in mate-terminal The problem:   I open a terminal (in Linux Mint, so mate-terminal) zsh is the shell Then I run tmux Edit a file with nano Scroll up and down that file with the cursor Issue: When scrolling down in nano, only the bottom half of the terminal window gets refreshed Issue: When scrolling up in nano, only the top half of the terminal windo gets refreshed   The complete nano view of file does not get refreshed in my terminal window when scrolling. Any tips?  Edit: my .tmux.conf  It seems that this line specifically is the culprit (as commenting it out fixes the problem):  set -g default-terminal "xterm-256color"   I'm pretty sure I added that line because I have issues even running nano during an SSH session.  Here is the full file:  set-option -g default-shell /bin/zsh  # Make sure tmux knows we're using 256 colours, for # correct colourised output set -g default-terminal "xterm-256color"  # The following were marked as "unknown", so # I do know what I'm doing wrong. #set -g mode-mouse on #setw -g mouse-select-window on #setw -g mouse-select-pane on  # Attempting to stop "alert" sound upon startup # but none of these are working... set-option bell-on-alert off set-option bell-action none set-option visual-bell off Asymptotic expansion for Fresnel Integrals Can you please guide me best way to retrieve all active picklist values of all recordtypes of object. Can it be possible through API or any Apex Script.  I want something like below and i will write them in my file:  Obect Name, RecordTypeName, PicklistValues   Thanks