How to select a font color to go on a very busy background?
Text not very readable on different image backgrounds
The series $\sum_{n=2}^{\infty }\frac{1}{\log (n!)}$ I'm trying to find out whether this Series $\sum_{n=2}^{\infty } a_{n}$ converges or not when $$a_{n}=\frac{1}{\log (n!)}$$  I tried couple of methods, among them: d'Alembert $\frac{a_{n+1}}{a_{n}}$,   Cauchy condensation test $\sum_{n=2}^{\infty } 2^{n}a_{2^n}$, and they both didn't work for me.  Edit: I can't use stirling, and integral.  Thank you Linked account +100 rep bonus also applied to account with 200 The following time-frequency correspondence illustration is in .    Also a gif animation version (if the animation doesn't show, please open the following image in a new window):    This picture has so much explanatory power, and I would like to replicate it in TikZ for future use.  Here is what I came up with:  \documentclass{minimal} \usepackage{tikz}  \begin{document} \begin{tikzpicture} [x={(1cm,0.5cm)},z={(0cm,0.5cm)},y={(1cm,-0.2cm)}] \draw[-&gt;,thick,blue!90] (0,6.5,0) -- (6.2,6.5,0) node[right] {Frequency}; \draw[-&gt;,thick,red!90] (0,0,0) -- (0,6.5,0) node[below] {Time}; \draw[-&gt;,thick] (0,0,0) -- (0,0,2) node[above] {Magnitude}; \foreach \x in {0.5,1.5,2.5,3.5,4.5,5.5}{   \draw[blue!50] (\x,0,0)   \foreach \y in {0,0.02,...,6.28}{     -- ({\x},{\y},{sin(\x*\y*(157))/sqrt(2*\x)})   };   \draw[blue!90, thick] (\x,6.5,0) -- (\x,6.5,1/\x); }  \end{tikzpicture} \end{document}   This is the output so far.    I have two questions:   How to produce that red superposed sine wave of all the blue sine waves? I don't know if there is a sum function or I have to use loop yet again? How to make the camera projection in TikZ more similar to the perspective in that wikipedia illustration?   Any suggestion and tweaking of the parameters I used in the sample drawing are welcome as well.  Thanks in advance!    Update 1:  Here is a new version using tikz, more readable than the first one. Yet the superposition of the sine waves are done manually...I still don't know how to use foreach to produce a sum.  \begin{tikzpicture}[x={(1cm,0.5cm)},z={(0cm,0.5cm)},y={(1cm,-0.2cm)}] \draw[-&gt;,thick,black!70] (0,6.5,0) -- (6.2,6.5,0) node[right] {Frequency}; \draw[-&gt;,thick,black!70] (0,0,0) -- (0,6.5,0) node[below right] {Time}; \draw[-&gt;,thick] (0,0,0) -- (0,0,2) node[above] {Magnitude}; \foreach \y in {0.5,1.5,...,5.5}{ \draw [cyan!50, domain=0:2*pi,samples=200,smooth]   plot (\y,\x, {sin(4*\y*\x r)/\y }); \draw[blue, ultra thick] (\y,6.5,0) -- (\y,6.5,1/\y); } \draw [red, thick, domain=0:2*pi,samples=200,smooth]  plot (0,\x, {sin(4*0.5*\x r)/0.5 + sin(4*1.5*\x r)/1.5 + sin(4*2.5*\x r)/2.5 + sin(4*3.5*\x r)/3.5 + sin(4*4.5*\x r)/4.5 + sin(4*5.5*\x r)/5.5} ); \end{tikzpicture}   The result is as follows: Transferring Fallout 4 Saves (PS4) What is the closed form of $\int e^{x^2} \, dx$? How can I do a closed form expansion of $\int e^{x^2} \, dx$? Please be specific as to which method I must use. Can't install php5 on Ubuntu 16.04 so I've recently installed my first Ubuntu, to be precise, a daily build of 16.04, on my new laptop since the hardware was to new for 15.10. Now I'm in the process of setting it up and I want to install php5 and some extensions.  When I execute sudo apt-get install libapache2-mod-php5 I get the error message that there is no installation package available for php5 and libapache2.  The error message is in german, so I doubt that this would help, but anyways here it is. :-) E: For package »libapache2-mod-php5« existiert kein Installationskandidat.  Do I need to add a apt-get repository? Or what am I doing wrong? Complete induction proof that every $n > 1$ can be written as a product of primes Excel macro to highlight all cells that match value in current cell $\lfloor x\rfloor + \lfloor y\rfloor \leq \lfloor x+y\rfloor$ for every pair of numbers of $x$ and $y$ Give a convincing argument that $\lfloor x\rfloor + \lfloor y\rfloor \leq \lfloor x+y\rfloor$ for every pair of numbers $x$ and $y$. Could someone please explain how to prove this? I attempted to say that the largest values that could be added to $x$ and $y$ is $0.99$ and that doing so still made $\lfloor x\rfloor + \lfloor y\rfloor &lt; \lfloor x+y\rfloor$. However, my answer was not accepted. Unable to correct problems, you have held broken packages After upgrading from 10.04 to 12.04 I am trying to install different packages.  For instance ia32-libs and skype (4.0).  When trying to install these, I am getting the 'Unable to correct problems, you have held broken packages' error message.   Output of commands:  sudo apt-get install -f Reading package lists... Done Building dependency tree        Reading state information... Done 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded.   After running this:  sudo dpkg --configure -a foo@foo:~$ sudo apt-get install -f Reading package lists... Done Building dependency tree        Reading state information... Done 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. CS0120: An object reference is required for the nonstatic field, method, or property 'foo' Consider:  namespace WindowsApplication1 {     public partial class Form1 : Form     {         public Form1()         {             InitializeComponent();         }          private void button1_Click(object sender, EventArgs e)         {             //int[] val = { 0, 0};             int val;             if (textBox1.Text == "")             {                 MessageBox.Show("Input any no");             }             else             {                 val = Convert.ToInt32(textBox1.Text);                 Thread ot1 = new Thread(new ParameterizedThreadStart(SumData));                 ot1.Start(val);             }         }          private static void ReadData(object state)         {             System.Windows.Forms.Application.Run();         }          void setTextboxText(int result)         {             if (this.InvokeRequired)             {                 this.Invoke(new IntDelegate(SetTextboxTextSafe), new object[] { result });             }             else             {                 SetTextboxTextSafe(result);             }         }          void SetTextboxTextSafe(int result)         {             label1.Text = result.ToString();         }          private static void SumData(object state)         {             int result;             //int[] icount = (int[])state;             int icount = (int)state;              for (int i = icount; i &gt; 0; i--)             {                 result += i;                 System.Threading.Thread.Sleep(1000);             }             setTextboxText(result);         }          delegate void IntDelegate(int result);          private void button2_Click(object sender, EventArgs e)         {             Application.Exit();         }     } }   Why is this error occurring?     An object reference is required for the nonstatic field, method, or property 'WindowsApplication1.Form1.setTextboxText(int) On a plane, there are two congruent circles, $O_1, O_2$. One line intersects $O _1$ and A and B and $O_2$ at C and D such that AB=BC=CD=14. Another line intersects $O _1$ and E and F and $O_2$ at G and H such that EF=FG=GH=6. What is the radius of the circle.   The configuration was hard to draw. . You can reflect the lines over the centers, but it is still the same. Cleary, $\widehat{AB}=\widehat{CD}$ and $\widehat{EF}=\widehat{GH}$. I have no idea how to proceed. 'this is' or 'these are'? I have the following sentences.        This is our first arrival in the UK.   These are our first arrival in the UK.      I am not sure, which sentence is correct? Can I configure Ubuntu to never install a specific package even if it is required by another package I install? How can I install applications like Google Chrome on Ubuntu?  Are there any commands to install an application?    Index of answers: I have a song that Windows can't play because there is a question mark in the name of the file.  "Where Have All the Cowboys Gone?.ogg" // as an example   So I try to rename it and Windows complains whether I try it in Explorer or from command prompt.  Error I get when trying to copy, rename, or move is:     The Filename, directory name, or volume label syntax is incorrect   Is there a Windows way to force a rename in this case?   Update  I'll keep an eye on this question, but after 13 answers and many attempts (aside form 3rd party solutions) it seems that Windows can't do this (or at least my windows can't, no short names). So I'm accepting the answer which was my original solution anyway of using Linux. It would be nice to see Windows handle this somehow, so don't stop just because I've accepted this answer, the question still stands! How to change Gnome-Terminal title? I have several instances of terminals running in my working environment, what I would like is to set a specific title for each one, in order to have a clear idea what purpose the specific terminal serves i.e. Apache, editing_ini, postgres etc...  Of course from the command line. Do I need to apply for a visa to enter South Korea as a tourist on an Australian passport? Do I need to apply for a visa to enter South Korea as a tourist for a few days when travelling on an Australian passport? Or will I be able to get one on arrival? Who were the hooded people in The Rise of Skywalker? Near the end of Star Wars Episode IX: The Rise of Skywalker there are several hooded figures that are shown chanting stuff (in Sith?) and just hanging out with Palpatine. Who were these people? Dynamic dispatch and binding