Infinite sums of reciprocal power: $\sum\frac1{n^{2}}$ over odd integers The infinite series I need to solve is  $$\sum_{n=1,3,5...}^{\infty}\frac{1}{n^{2}}$$  and because the point of interest lies in the value of odd n,   the infinite series can be expressed as  $$\sum_{n=1}^{\infty}\frac{1}{(2n-1)^{2}}$$  This came up in a quantum mechanics problem involving the expectation value of the Hamiltonian.   Is there a good idea to verify the solution is indeed $$\frac{\pi^{2}}{8}$$ or is this something with which I must refer to a math table? Any good ideas would be helpful.
how can we show $\frac{\pi^2}{8} = 1 + \frac1{3^2} +\frac1{5^2} + \frac1{7^2} + …$? Let $f(x) = \frac4\pi \cdot (\sin x + \frac13 \sin (3x) + \frac15 \sin (5x) + \dots)$. If for $x=\frac\pi2$, we have $$f(x) = \frac{4}{\pi} ( 1 - \frac13 +\frac15 - \frac17 + \dots) = 1$$ then obviously :  $$ 1 - \frac13 +\frac15 - \frac17 + \dots=\frac{\pi}{4}$$ Now how can we prove that: $$\frac{\pi^2}{8} = 1 + \frac1{3^2} +\frac1{5^2} + \frac1{7^2} + \dots$$
How can I change my Minecraft username? I am using NCR18650 batteries with the tp4056 charging module.  When I connect the charger at 2.8&nbsp;V (fully discharged), the tp4056 IC is getting hot at a 1&nbsp;A charge current. Is it OK to reduce this charge current by changing the charge current selecting resistor? So the tp4056 IC heat would be reduced, wouldn't it?  For my application I don't need a quick charge. With a low current, charging time will be increased, but it does not matter for my application.  Will reducing the charge current affect the battery's lifetime? Or will it damage the battery?  I am trying to reduce the heat of the tp4056 IC by reducing the charging current, maybe from 1&nbsp;A to 500&nbsp;mA or less. The 18650 battery never heats for a 1&nbsp;A charge current. Partial Fraction Decomposition? Is $\mathbb{Z}[x]/(x^2+1)$ isomorphic to $\mathbb{Z}[i]$? So I defeated a Team Valor gym earlier today using my highest cp Pidgeot. It turned grey, so I went to claim it with that Pidgeot, except it wouldn't let me. What determines which pokemon you are allowed to place at a gym? Is it just those who have participated in a battle can't? Is there a time period for it? What about training at a friendly gym? Does this impact anything? Disable Tabbing in Firefox 3, Chrome 4 and IE8? Tabbed Browsing may be nice for reading news sites in one sitting, but I still think they are a usability nightmare if you need to keep the page open for a long time, e.g., for testing it while developing or as a reference for something.  The nice thing about the Windows Taskbar (before Windows 7 ) is that it's easy and natural to remember where things are. I can have 30 tabs in it, but because I know what each one is, I know where I have to click to open something - it's the most efficient way of working I've encountered so far.  Sadly, Tabbed Browsing completely breaks this efficiency because it is not possible to specifically click on a tab in the taskbar, I have to click on the browser, search the tab and then click it, which means at least 2 clicks.  Long story, short question: Is there an extension or setting to completely disable tabbing on Firefox 3, Chrome 4 and Internet Explorer 8 and have one Window/Taskbar entry for each open page? I know that in Chrome I can just drag a tab out of a Window to create a new one, but if there is a general way, that would be easier.  Edit: IE8 conveniently offers an option for that: Tools - Options - Tabs - Enable Tabbed Browsing. Can I combine extension tubes and macro filters at once to give better macros? Stars and pins are retained when chat messages are moved! In the chat system, stars and pins are retained when moved to another room. This is probably a bug. Here a picture:  When a room owner pins a message, they can move it to another room, and they cannot unpin it. However when a starred message is moved to another room, the user can unstar it in the room it was moved to. Conversation source:  * is irrelevant and does not have to do with this bug report. Many thanks to . The probability that an initial quantum state $|\psi_i\rangle$ becomes the final quantum state $|\psi_f\rangle$ is given by  \begin{eqnarray} P(i \rightarrow f) &amp;=&amp; |\langle\psi_f|\psi_i\rangle|^2 \tag{1}\\ &amp;=&amp; \langle\psi_f|\psi_i\rangle^*\langle\psi_f|\psi_i\rangle \\ &amp;=&amp; \langle\psi_i|\psi_f\rangle\langle\psi_f|\psi_i\rangle. \end{eqnarray}  Equation (1) seems to show that the probability for the transition ($i\rightarrow f$) can be interpreted as the system both moving forward in time ($i\rightarrow f$) with amplitude $\langle\psi_f|\psi_i\rangle$ and backward in time ($f\rightarrow i$) with amplitude $\langle\psi_i|\psi_f\rangle$ simultaneously.  Does this reasoning help to explain the Born rule? (Is it like the Transactional Interpretation of QM?)  I guess we must experience the macroscopic direction of time ($i\rightarrow f$) in accord with increasing entropy in an expanding universe whereas microscopically QM works both forwards and backwards in time.  Addition  This is an improved version of the argument including time-evolution operators.  The probability that an initial quantum state $|\psi_i\rangle$ evolves to become the final quantum state $|\psi_f\rangle$ is given by  \begin{eqnarray} P_{i \rightarrow f} &amp;=&amp; |\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle|^2 \tag{2}\\ &amp;=&amp; \langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle^*\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle \\ &amp;=&amp; \langle\psi_i|U^\dagger_{i \rightarrow f}|\psi_f\rangle\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle \\ &amp;=&amp; \langle\psi_i|U_{f \rightarrow i}|\psi_f\rangle\langle\psi_f|U_{i \rightarrow f}|\psi_i\rangle \end{eqnarray} where $U_{i \rightarrow f}$ is the forward-time evolution operator and $U_{f \rightarrow i}=U^\dagger_{i \rightarrow f}$ is the corresponding backward-time evolution operator.  Equation (2) seems to show that the probability $P_{i\rightarrow j}$ can be interpreted as the system first evolving forwards in time and then evolving backwards in time.  Perhaps this is an example of Murray Gell-Mann's  that "Everything not forbidden is compulsory"? At the quantum level, below observable probabilities, there is nothing to stop time flowing both forwards and backwards. In many scientific papers, the article before the word "Equation" is omitted. Is there any grammar statement behind this? For instance:     "Equation (8) contains various approximations, and we have to check their consistency." Can I visit other Schengen countries on a long-term Schengen visa or residency permit? I have requested for a French long term (Type D) student visa from a consulate in India. It's a national Type D visa and not a Schengen visa. My question is that can we still go to other Schengen countries on this visa? If not then can we get a Schengen visa from a consulate in France? Log form of variables and descriptive measures Because of larger values of variables, I did a log- transformation in my dataset. Now I want to give a descriptive table regarding my variables like mean, max, min, median, skewness, kurtosis. Can I give standard measures like arithmetic mean or should I give geometric mean? In terms of skewness or kurtosis, do I need a transformation? Zero-dimensional ideals and finite-dimensional algebras I encountered in the literature the term zero-dimensional ideal, however I can not find a relevant definition anywhere in Atiyah-MacDonald or Matsumura. In fact, I encounted the statement:      $I$ is a zero-dimensional ideal of $k[x_1,\dots,x_n]$ $\Leftrightarrow$ $k[x_1,\dots,x_n]/I$ is a finite-dimensional $k$-vector space, where we assume that $k$ is algebraically closed.   Question 1: What is the definition of the dimension of an ideal?  Question 2: How do we prove the above statement?  Question 3: Is $k$ being algebraically closed a necessary condition for the statement to be true? how can I find sum of series $ \sum_{n=1}^{\infty} (n\cdot \ln \frac{2n+1}{2n-1} - 1) $? It is so weird for me because I put this to Mathematica and it tells me that sum does not converge...  Let consider sum no to infinity, but to n  $$ \sum_{k=1}^{n} (k\cdot \ln \frac{2k+1}{2k-1} - 1) =$$ $$ ln \frac{3}{1}\cdot \left(\frac{5}{3}\right)^2 \cdot...\cdot \left(\frac{2n+1}{2n-1}\right)^n - n = ln \frac{1}{1}\cdot \frac{1}{3}\cdot \frac{1}{5}\cdot ... \frac{1}{2n-1} - n $$ but $$ n = ln e^n $$ so it will be $$ln\frac{1}{e^n} \cdot \frac{1}{1}\cdot \frac{1}{3}\cdot \frac{1}{5}\cdot ... \frac{1}{2n-1}$$  So the limit of it is $-\infty$ Have I done this well or I missed sth? I have the following data structure (a list of lists)  [  ['4', '21', '1', '14', '2008-10-24 15:42:58'],   ['3', '22', '4', '2somename', '2008-10-24 15:22:03'],   ['5', '21', '3', '19', '2008-10-24 15:45:45'],   ['6', '21', '1', '1somename', '2008-10-24 15:45:49'],   ['7', '22', '3', '2somename', '2008-10-24 15:45:51'] ]   I would like to be able to   Use a function to reorder the list so that I can group by each item in the list. For example I'd like to be able to group by the second column (so that all the 21's are together)  Use a function to only display certain values from each inner list. For example i'd like to reduce this list to only contain the 4th field value of '2somename'    so the list would look like this   [      ['3', '22', '4', '2somename', '2008-10-24 15:22:03'],       ['7', '22', '3', '2somename', '2008-10-24 15:45:51'] ] This is a follow-up to : ulimit and cgroups are not user friendly, and besides, wouldn't work with applications that spawn separate processes, such as Chrome/Chromium for each new (group of) tabs.  The simple and effective solution, used by Windows 7 actually, is to warn the user that the OS is running low on memory. This simple warning pop-up has prevented me from having any low-memory-caused system freeze in Windows, while I kept running into them on Ubuntu distros that I was testing live (where the RAM-mounted disk would eat up 2GB alone).  So, is there some way to automatically warn the user that the available RAM is nearing zero, without the user having to keep an eye on some memory monitoring gadget? Surely  could be configured to do that? Why do I only have 30V going to my light fixture In the textbook, it said a wave in the form $y(x, t) = A\cos(\omega t + \beta x + \varphi)$ propagates along negative $x$ direction and $y(x, t) = A\cos(\omega t - \beta x + \varphi)$ propagates along positive $x$ direction. This statement looks really confusing because when it says the wave is propagating along $\pm$ x direction, to my understanding, we can drop the time term and ignore the initial phase $\varphi$ while analyzing the direction, i.e. $y(x, 0) = A\cos(\pm\beta x)$, however, because of the symmetry of the cosine function, $\cos(\beta x)\equiv \cos(-\beta x)$, so how can we determine the direction of propagation from that?  I know my reasoning must be incorrect but I don't know how to determine the direction. So if we don't go over the math, how to figure out the direction of propagation from the physical point of view? Why $-\beta x$ corresponding to the propagation on positive x direction but not the opposite? When I try to move, rename, or delete a file named ␀in Finder, I get this error message:     The operation can't be completed, because an unexpected error occured (error code -50).   With mv, rm or find -delete in Terminal   rm: /Library/␀: Invalid argument  mv: rename /Library/␀ to /tmp/␀: Invalid argument  find: -delete: unlink(/Library/␀): Invalid argument   Can I move or delete this file in Mac OS X or do I have to use another operating system?  I even tried creating a file with this name in /tmp and it has not gone away since 8 reboots. A friend asked me last week how to enumerate or list all variables within a program/function/etc. for the purposes of debugging (essentially getting a snapshot of everything so you can see what variables are set to, or if they are set at all). I looked around a bit and found a relatively good way for Python:   #!/usr/bin/python                                                                                                                                                                                                                            foo1 = "Hello world" foo2 = "bar" foo3 = {"1":"a",         "2":"b"} foo4 = "1+1"  for name in dir():     myvalue = eval(name)     print name, "is", type(name), "and is equal to ", myvalue   which will output something like:   __builtins__ is &lt;type 'str'&gt; and is equal to  &lt;module '__builtin__' (built-in)&gt; __doc__ is &lt;type 'str'&gt; and is equal to  None __file__ is &lt;type 'str'&gt; and is equal to  ./foo.py __name__ is &lt;type 'str'&gt; and is equal to  __main__ foo1 is &lt;type 'str'&gt; and is equal to  Hello world foo2 is &lt;type 'str'&gt; and is equal to  bar foo3 is &lt;type 'str'&gt; and is equal to  {'1': 'a', '2': 'b'} foo4 is &lt;type 'str'&gt; and is equal to  1+1   I have so far found a partial way in PHP (courtesy of ) but it only lists all variables and their types, not the contents:   &lt;?php // create a few variables $bar = 'foo'; $foo ='bar'; // create a new array object $arrayObj = new ArrayObject(get_defined_vars()); // loop over the array object and echo variables and values for($iterator = $arrayObj-&gt;getIterator(); $iterator-&gt;valid(); $iterator-&gt;next())         {         echo $iterator-&gt;key() . ' =&gt; ' . $iterator-&gt;current() . '&lt;br /&gt;';         } ?&gt;   So I put it to you: how do you list all variables and their contents in your favorite language?    Edit by : I propose this question follows the spirit of a little "". If you do not agree, just edit and remove the tag and the link.