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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. What issues / pitfalls must be considered when overriding equals and hashCode? fetching list of friends using facebook 2.0 How do I set read/write permissions my hard drives? I consistently have more than one terminal open. Anywhere from two to ten, doing various bits and bobs. Now let's say I restart and open up another set of terminals. Some remember certain things, some forget.  I want a history that:   Remembers everything from every terminal Is instantly accessible from every terminal (eg if I ls in one, switch to another already-running terminal and then press up, ls shows up) Doesn't forget command if there are spaces at the front of the command.   Anything I can do to make bash work more like that? How to clear Chocolatey cache in the free version? Installation-file-cleaning is currently only available in  () and .  The package installation files can easily take gigabytes of storage space and to my knowledge they only increase in size as the user updates packages and installs new software, with only your hard drive capacity as the upper limit.   I don't know if Chocolatey stores cache in one central location — something is saved to ~\AppData\Local\Temp but the majority is located in the lib* folders in ~\ProgramData\chocolatey. Also the log files in the logs folder total to over 100 MB on my system. Is there any way to fully clear Chocolatey cache without manually deleting directories? How do Idle games progress while not active? How to recover a deleted file from /data partition? I factory reset my rooted Android M phone but now I realize that there was a file in the /data directory which I need.  Is it possible to create an image (.img or something similar) of the /data partition so that I could run some recovery software for ext4 on that image? I have a little knowledge about ADB. How to make the term "page" show in the Glossary list? Find the limit of $\frac{n^{x}}{(1 + x)^{n}}$ Let $x&gt;0$. Find $$\lim_{n\to\infty}\frac{n^{x}}{(1 + x)^{n}}.$$   What I was thinking involved an inequality, by reason of considering the fact that  $$(1 + x)^{n} &gt; C(n, k) x^{k}$$ for some $k$ such that $n&gt;k$. Expanding the binomial coefficient and taking the $n$ common from each bracket, we have something like $$(1 + x)^{n} &gt; C(n, k) x^{n} &gt; \frac {x^{k}n^{k}}{k! 2^{k}}.$$ But now, I am totally clueless. By intuition, I know that the limit has to be zero, so I just want to prove using the concept of inequalities that what I thought was the right approach (being the sandwich theorem). Any help would be appreciated! How can I change GNOME Shell theme? I've installed the GNOME Shell in Ubuntu 12.04 and I want to change the "Shell Theme", But it doesn't allow me to do that.  The alert says something like: "The GNOME-Shell extension is not activated", and I want to install it with the console, But it returns me an error.  How can I solve this problem? Can you stay in Turkey for over 90 days within the 180 day visa period but split over different visits? The Turkish e-visa is valid for a period of 180 days and allows you to stay for a maximum of 90 days. Am I allowed to stay for say 50 days, then another 50 days, over two different visits within the 180 day period?  I'm looking at the e-visa website  and it doesn't seem to be too clear on this. Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx$ I am trying to find a closed form for  $$\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx = 0.094561677526995723016 \cdots$$  It seems that the answer is $$\frac{\pi^2}{12}\left( 1-\sqrt{3}\right)+\log(2) \log \left(1+\sqrt{3} \right)$$  Mathematica is unable to give a closed form for the indefinite integral.  How can we prove this result? Please help me.  EDIT  Apart from this result, the following equalities are also known to exist: $$\begin{align*} \int_0^1 \frac{\log \left( 1+x^{4+\sqrt{15}}\right)}{1+x}\mathrm dx &amp;=\frac{\pi^2}{12} \left( 2-\sqrt{15}\right)+\log \left( \frac{1+\sqrt{5}}{2}\right)\log \left(2+\sqrt{3} \right) \\ &amp;\quad +\log(2)\log\left( \sqrt{3}+\sqrt{5}\right) \\ \int_0^1 \frac{\log \left( 1+x^{6+\sqrt{35}}\right)}{1+x}\mathrm dx &amp;= \frac{\pi^2}{12} \left( 3-\sqrt{35}\right)+\log \left(\frac{1+\sqrt{5}}{2} \right)\log \left(8+3\sqrt{7} \right) \\ &amp;\quad +\log(2) \log \left( \sqrt{5}+\sqrt{7}\right) \end{align*}$$  Please take a look . How to upload files to server using JSP/Servlet? How to twist radial vector to make spirals? Add Word Limit to Posts Methods to compute $\sum_{k=1}^nk^p$ without Faulhaber's formula As far as every question I've seen concerning "what is $\sum_{k=1}^nk^p$" is always answered with "Faulhaber's formula" and that is just about the only answer.  In an attempt to make more interesting answers, I ask that this question concern the problem of "Methods to compute $\sum_{k=1}^nk^p$ without Faulhaber's formula for fixed $p\in\mathbb N$".  I've even checked this  without finding what I want.  Rule #1: Any method to compute the sum in question for arbitrary $p$ is good, either recursively or in some manner that is not in itself a closed form solution.  Even algorithms will suffice.  Rule #2: I don't want answers confined to "only some values of $p$".  (A good challenge I have on the side is a generalized geometric proof, as that I have not yet seen)  Exception: If your answer does not generalize to arbitrary $p$, but it still generalizes to an infinite amount of special $p$'s, that is acceptable.  Preferably, the method is to be easily applied, unique, and interesting.  To start us off, I have given my answer below and I hope you all enjoy. Unknown Syntax Error in Code So, I am following a , and it is using Python code to create an audio visualizer in Blender. Now, I've always been a coding geek, but I don't actually know Python. I've taken a few shots at it but overall, I don't know Python, however, I can read and understand the code because I know basic programming concepts. Now, I'm following this tutorial as closely as I can (the tutorial does not provide code to copy and paste) and I am getting a syntax error. As far as I can tell, there is no syntax error. The line of code (specified by the error message) appears perfect. This line of code is supposed to set the 3D cursor to the active object's origin. Here's the catch: if I run that exact same line of code in the console, it executes just fine. So here's the error message I get:  File "C:\Me\Blender\Blender Projects\audio visualisation.blend\AudioVisualizer", line 17 bpy.context.scene.cursor_location = bpy.context.active_object.location    ^ SyntaxError: invalid syntax location: &lt;unknown location&gt;:-1   It tells me to look in the console for error message, but the console doesn't output anything.    Thanks!  EDIT: Alright. I've tried all the examples and all the procedures that have been suggested by you guys, but it keeps feeding me errors. According to the console, one of the problems it is spitting out is in C:\Program Files\Blender Foundation\Blender\2.74\scripts\modules\bpy\ops.py. This makes me think there is a bug in the API that is causing one of the many errors. EDIT: Here are the error messages I get for my code. Original Code:  Error:   File "C:\Gannon\Blender\Blender Projects\audio visualisation.blend\Audi o Visualizer", line 35     bpy.ops.graph.sound_bake(filepath="C:\Gannon\Blender\Sounds\Music\Batty McFa dden.mp3", low=i*step high=i*step+step                           ^ SyntaxError: invalid syntax    gandalf3's Code and zeffii's Code are essentially the same with alternative formatting which changes the line numbers where the error occurs.  Error: Unsupported audio format  Traceback (most recent call last):   File "C:\Gannon\Blender\Blender Projects\audio visualisation.blend\Audio Visua lizer", line 42, in &lt;module&gt;   File "C:\Gannon\Blender\Blender Projects\audio visualisation.blend\Audio Visua lizer", line 35, in spiral   File "C:\Program Files\Blender Foundation\Blender\2.74\scripts\modules\bpy\ops .py", line 189, in __call__     ret = op_call(self.idname_py(), None, kw) RuntimeError: Error: Unsupported audio format What does 'per unit frequency' refer to? We were discussing Kirchoff's theorem on blackbody radiation in class today and the equation was: $$e_{f}= J(f,T)$$  $e_f$ was defined as power per unit area per unit frequency. What does per unit frequency refer to here? I know what per unit time means but don't understand 'per unit frequency'. Can I deduct a contribution to a traditional IRA if I am also contributing to a 401(k) in the same tax year?