How to prove $$1+\cfrac{1}{1+\cfrac{e^{-2\pi}}{1+\cfrac{e^{-4\pi}}{1+\cfrac{e^{-6\pi}}{\cdots}}}} = \left(\sqrt{5\phi}-\phi\right) e^{2\pi/5}.$$ i dont know how to do it. like if there were repeating terms we put it to be $z$ and then again $1/z$ and solve the quadratic  but how to go for this. Note that I am interested if there is a  way using elementary knowledge as I am only in grade $12$. Thanks!
I found this problem on a well-known problem solving website. It is apparently from Ramanujan.     With   $$\LARGE{a = \frac{1}{1 + \frac{e^{-2\pi}}{1 + \frac{e^{-4\pi}}{1 + \ddots}}}},$$   what is $a$?   The basic methods I know for questions of this type were not helpful to me.  I do not require a full solution, but feel free to post one if you would like. Hints or lecture notes on how to solve problems of this nature would be appreciated.   Keep in mind that I am at an undergraduate level in mathematics.
How do I create a decimal field in Access with Alter Table? How do I support scrolling and lock/freeze the header row of a pageBlockTable? Binomial Distribution Problem - Airline Overbooking How do I get an animated gif to work in WPF? How to change the list of file types which Safari will open immediately after download? How to align two images side by side if one label has more lines I have two same sized images and want them side by side with a caption under them.  My code looks like this:   \begin{figure}[H] \centering \begin{minipage}{.5\textwidth}   \centering     \includegraphics[width=\textwidth]{3}     \caption{label1}   \label{fig:test1} \end{minipage}% \begin{minipage}{.5\textwidth}   \centering     \includegraphics[width=\textwidth]{4}     \caption{label2}   \label{fig:test2} \end{minipage} \end{figure}   which works fine until one of the labels has 2 lines, than it looks like this:    and I woud like to prevent it to look something like this:    How can it be done?  Thanks This question is about cosmology and general relativity. I understand the difference between the universe and the observable universe. What I am not really clear about is what is meant when I read that the universe is infinite.   Does it have infinite mass or is it dishomogeneous?  How can the universe transition from being finite near the big bang and infinite 14 billion years later? Or would an infinite universe not necessarily have a big bang at all? Prove that if for every $x \in \mathbb{R}^N$ $Ax=Bx$ then $A=B$ Cannot delete files, nothing seems to work I am on Windows 7.  I have some files on my computer leftover from an old version of Photoshop.  (I think... I don't remember exactly where they came from, but they seem to be Adobe files.)  There are 124 of them in a folder, which I have named "KILL."  (I don't remember what the original name of this folder is, it's been on my comptuer a long time.)  These files seem to be absolutely undeletable, as far as I can tell.  I can move them around and change the name of the folder, but deleting them is out of the question.  I also can't move them onto a different hard drive or flash drive, I've tried that.  So, let me list the things I've tried first:   A whole bunch of different "unlocking programs."  Including: IObit Unlocker, Lock Hunter, FileAssassin, and Project Kill box.  What I use LockHunter I am told: "No processes locking this file or folder have been found."  All of the other programs I've tried give me similar messages, telling me that nothing is locking the files.  Yet when I try to delete the files using LockHunter, I am simply told "Cannot delete x."  No reason is given. Tried using cmd.exe.  Even when I run a command window under administrator, it won't even let me access the KILL folder, much less delete anything.  I also tried killing explorer.exe from the task manager, and then using the cmd window.  Still can't even access that folder. Tried using safe mode in combination with these other options, doesn't change a thing. Tried changing permissions of the files using the security tab under properties.  It simply says "not inherited."  It will not allow me to change a thing. I am definitely the administrator of this computer. Tried Disk Cleanup and Defrag.  I suspected that some files must be damaged or something, but Disk Cleanup wasn't able to find any errors.  Defrag didn't help.   So, let me just say that I am not a computer expert.  But based on all the things I've tried, I suspect that this might be some hardware problem?  It seems strange that these files don't seem to be used by any processes, and yet are absolutely undeletable no matter what I do.  Is there anything else I can try?  Has anyone had any similar experience, and have any idea why this might have happened?  EDIT: Interesting recent development.  As I have toyed with moving the folder around, I moved it to the Program Files folder.  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Additionally, I've been getting rain on a UV filter I have on the front of my lenses, simply to prevent rain from getting on the front lens element itself. However, once a storm really gets going, so does the rain. Is there a simple solution to keep rain off the front of the lens? In my book, the following are described to be the steps that one must follow if he wishes to find the direction of static friction force acting on an object:  &quot;(1) Draw the F.B.D(free body diagram) of the object in question with respect to the other object on which it is kept. (2) Include the pseudo force also, if the contact surface is accelerating. (3) Decide the direction of the resultant force and resolve this force into two components; one along the surface of contact and the other along the normal to this surface. (4) The direction of static friction is opposite to the component of the result force along the contact surface.&quot;  Firstly, are these steps correct? Will I always obtain the correct answer by following these steps? Secondly, consider the following problem:  In questions like these, when I try to find the resultant force and then assign the direction to static friction, I find that my answer is incorrect! Are there some other steps to follow when I know that the object is definitely accelerating in one direction or another? Edit 1: I approached the above question by doing exactly what my book suggests; first, I identified all the forces acting on the smaller block, being the normal force applied by the block A and its weight mg downward. Adding these two forces, I get another force that actually acts on the body like so:  I wonder if I'm not supposed to take either of the contact forces into consideration when drawing the F.B.D to figure out the direction of static friction acting on the body. Also, are there other steps for finding the direction of kinetic friction? Edit 2: From Judge's answer, I have been able to figure out my mistake, it being that I completely missed step (3) and didn't resolve the resultant force into components! Sheesh. Sorry! Much thanks in advance! :) Regards. Counting area percentage of overlapping polygons I have a set of points (meteorological stations) with 5km buffers around them. Additionally, I have a set of polygons (municipalities) in which those stations are located.  I would like to have data on how much of the buffer is filled by each polygon but I was not able to find a function or a way to do it. So for example for point A I would have that in the buffer of 5km there is 80% of polygon B and 20% of polygon C as the output.  Edit:  The anwsers provided do anwser this question in a setting in which I know what polygons and buffers I need to have in the code or when I can do it manualy. Problem is that in my setting with 1 shapefile having 500 polygons and one vector layer having 13000 buffers I am unable to do this manually and I am asking if there is an easier way to do it. I tried implementing the approaches in the questions but I was unable to use them on my example in big scale. Moreover I had problems with coding that I also noted in comments down below.   Main difference to the other similar question provided is that I have 2 vector layers (Buffer with 500 buffers and Municipalities a shape file with 500 different polygons). Previous questions were analyzing this question in situations in which I know precisely the polygons and buffers analyzed, at least that is how I understood them, so after trying to use those approaches on my datasets I was not able to replicate them.  Another way of approaching this problem for me is to use join attributes by location -one-to-many and then getting averages of the duplicate results, only problem is that then I take average of polygons instead of their weights corresponding to % of intersection with the buffer. In my case I do not know what polygons of the shapefile are going to intersect what buffers, or even if the buffer is going to be intersected by 1,2 or more polygons Find the limit $$\lim_{x\to 0^{+}}\dfrac{\sin{(\tan{x})}-\tan{(\sin{x})}}{x^7}$$  My attempt: Since $$\sin{x}=x-\dfrac{1}{3!}x^3+\dfrac{1}{5!}x^5-\dfrac{1}{7!}x^7+o(x^7)$$ $$\tan{x}=x+\dfrac{1}{3}x^3+\dfrac{2}{15}x^5+\dfrac{1}{63}x^7+o(x^3)$$ So $$\sin{(\tan{x})}=\tan{x}-\dfrac{1}{3!}(\tan{x})^3+\dfrac{1}{5!}(\tan{x})^5-\dfrac{1}{7!}(\tan{x})^7+o(x^7)$$ Though this method might solve, I think this problem has nicer methods. Thanks. Open more than one instance of Windows 10 settings How can I open more than one instance of the Windows settings?  It wasn't a problem with the old style, but with the new one I can't seem to open multiple settings simultaneously and side by side.  Example of the new settings window: Overriding !important style Title pretty much sums it up.  The external style sheet has the following code:  td.EvenRow a {   display: none !important; }   I have tried using:  element.style.display = "inline";   and  element.style.display = "inline !important";   but neither works. Is it possible to override an !important style using javascript.  This is for a greasemonkey extension, if that makes a difference. Symbolizing layer with graduated colors by date/time field? I've got a layer of polygons where each has a date associated with it.  I want to color the polygons with graduated colors, but I want the colors to start with the newest polygons and fade to the oldest polygons.  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