In math lesson, my teacher told me that Euler once used a very delicate method to calculate $\displaystyle S_{n}=\sum_{i=1}^{n}\frac{1}{i^{2}}$ and wrote a paper about it.  I wonder how to calculate this. I need a precise answer, not the answer that it's less than a number such as 2.
As I have heard people did not trust Euler when he first discovered the formula (solution of the ) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler and he gave other proofs.   I believe many of you know some nice proofs of this, can you please share it with us?
Ubuntu doesn't boot! Travelling through the Schengen area with a Refugee Travel Document issued by the USA I just want to double check to make sure I won't run into any issues at the border of entry.  I'm a permanent resident of the USA with a green card and I have a travel document issued by the USA (I-571). Will I have any issues traveling around in the Schegen countries?  Do I need a visa? Do I have to pay any fees? Show that $\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y}) = \sum_{i=1}^n (x_i - \bar{x})(y_i)$. Show that $\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y}) = \sum_{i=1}^{n} (x_i - \bar{x})(y_i)$. I fell into this hole where I keep finding that $$\sum_{i=1}^n (x_i - \bar{x}) = \sum_{i=1}^n x_i - \sum_{i=1}^n \bar{x} = n\bar{x} - n\bar{x} = 0$$ This seems true, but I know that I am missing something, since this implies that covariance is always zero, which is obviously not true. Does the mass of a battery change when charged/discharged? If so, how much? Is it possible to detect it, or is it beyond any measurement?  I'd say there are two possible scenarios (depending on the battery type) and both seem interesting:   The battery reacts chemically with its environment. The battery doesn't exchange any matter with its environment except electrons. I suppose there should be some difference at least due to the principle of energy-matter equivalence, but the difference is most likely immeasurable. The only things I've found so far are a cart method (which ceased working in 1.6), a dispenser method (utilizing the decay on dropped items, but this requires occasional feeding), and a binary counter (which won't work for a single input but rather a fluctuating one).  I want to have a circuit that when I apply an input signal, the output signal doesn't change for 2 to 5 minutes.  When the input turns off, I again want a 2 to 5 minute delay.  I could accomplish this with (roughly) 240 to 600 redstone repeaters, but this is far in excess of being useful.  The only method I can think of, but would require a XOR gate (on the button-input and final output) to start and stop a clock attached to a binary counter.  While I can make room for a device of this size, I was hoping for something simpler. How to remove a filter that is an anonymous object? I am learning Linux and I was trying the gzip command. I tried it on a folder which has a hierarchy like  Personal/Folder1/file1.amr Personal/Folder2/file2.amr Personal/Folder3/file3.amr Personal/Folder4/file4.amr   I ran     "gzip -r Personal" and now its like  Personal/Folder1/file1.amr.gz Personal/Folder2/file2.amr.gz Personal/Folder3/file3.amr.gz Personal/Folder4/file4.amr.gz   How do I go back? How to clone ArrayList and also clone its contents? DOS commands for Epson dot matrix not working Our Epson dot matrix printer is not printing on the following DOS command.:  Type text.txt&gt;prn   But it does print perfectly on:  open TEXT.txt&gt;File&gt;Print    What is the issue here? I'd like to work on a bbcode filter for a php website. (I'm using cakephp, it would be a bbcode helper) I have some requirement.    Bbcodes can be nested. So something like that is valid.  [block]       [block]       [/block]       [block]           [block]           [/block]       [/block]   [/block]     Bbcodes can have 0 or more parameters.     Exemple:     [video: url="url", width="500", height="500"]Title[/video]   Bbcodes might have mutliple behaviours.     Let say, [url]text[/url] would be transformed to [url:url="text"]text[/url] or the video bbcode would be able to choose between youtube, dailymotion....  I think it cover most of my needs. I alreay done something with regex. But my biggest problem was to match parameters. In fact, I got nested bbcode to work and bbcode with 0 parameters. But when I added a regex match for parameters it didn't match nested bbcode correctly.   "\[($tag)(=.*)\"\](.*)\[\/\1\]"  // It wasn't .* but the non-gready matcher  I don't have the complete regex with me right now, But I had something that looked like that(above).  So is there a way to match bbcode efficiently with regex or something else.  The only thing I can think of is to use the visitor pattern and to split my text with each possible tags this way, I can have a bit more of control over my text parsing and I could probably validate my document so if the input text doesn't have valid bbcode. I could Notify the user with a error before saving anything.  I would use sablecc to create my text parser.   Any better idea? or anything that could lead to a efficient flexible bbcode parser?  Thank you and sorry for my bad english... Explicit bijection between $\Bbb R$ and permutations of $\Bbb N$ Definition of the Pauli group and the Clifford group There seem to be two definitions of the Pauli group. In Nielsen and Chuang, the Pauli group on 1 qubit is defined as \begin{align*} \mathcal{P}_1 = \{\pm I, \pm iI, \pm X, \pm iX, \pm Y, \pm iY, \pm Z, \pm iZ\} = \langle X, Y, Z\rangle. \end{align*} But in other references, such as ,  \begin{align*} \mathcal{P}_1 = \{\pm I, \pm X, \pm iY, \pm Z\} = \langle X, Z\rangle. \end{align*} The second definition makes more sense in the stabilizer formalism, since we usually only consider the action of Pauli strings of $X$ and $Z$. What bothers me is that by the second definition, the phase gate $S$, which is an element of the Clifford group, results in \begin{align*} SXS^\dagger = Y \notin \mathcal{P}_1. \end{align*} I know that the prefactor $\pm i$ somehow doesn't matter, but can someone justify it from a more mathematical/group-theoretical perspective? How do I log every command executed by a user? I want to monitor all user's activity in my server.  Even when the user executes a shell command from some editor like vim I want to see them in the log file.  I have checked the tool acct but it is not listing the complete commands. (Please correct me if I have missed some options which does already).  Which Linux tool I should be looking at to solve this problem? Here at the Genealogy and Family History Stack Exchange we see many questions about relationship mapping which is also known as:   relationship calculation kinship terminology etc   If you are just trying to understand how to calculate "cousins and removes" then  provides an excellent guide.  However, if you are seeking an English term for a more complicated relationship between two members of an extended family, rather than asking a question like:          site users are encouraged to first review the answers below.    Note: This Q&amp;A arises from  and is set to  to help make it canonical. In the recently launched trailer of Fantastic Beasts: The Crimes of Grindelwald, there are people Apparating inside Hogwarts grounds.                                  As these people later question Professor Dumbledore, it seemed they were Ministry officials.  Have we ever encountered such Apparition before? Or, was this anti-Apparation charm put into place in recent times? On September 29th, I created the  tag for  Today, while checking the tag description, I saw that the  seems to be .  Why is that? Why has the Community user "stolen" the spot as author of the tag? Is it true that the light from some galaxies will never reach us?   The explanation for that is that the Universe expanding faster than the speed of light. But, if the speed of light is constant in all reference frames, then in the reference frame of a galaxy, light must move at the speed of light and thus it must reach us (?).   Could you also explain  how is it possible that the Universe is expanding faster than the speed of light if an object can never be observed to move faster than light? (without Doppler effects). So I'm pretty sure there's at least 2, one for both X &amp; Y.   Much like other iterations of the other Pokemon games, are there other Legendaries? How many legendaries, and what are their names? How to prove  $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$ How do you maximize the cycle count of your battery?  There seems to be a debate between:   Leave plugged in 24/7 but do a full cycle once a month Charge to ~100%, drain to ~10%, repeat.   NOTE: Taking your   Which method is better and why? How much of a difference does it really make?  Here are some of the better sources I've found about battery behavior so far. Even still I feel they don't conclusively answer the question above. See my elaborations below:           Elaborating on Option 1  Now IF Apple engineers were smart and optimized for AC draw when plugged in to not use up your charge cycles, then it seem logical to assume that you are using effectively none of your 1000 charge cycles while plugged in. If that's the case, it would seem that the LiPo drop from holding at 100% would be vastly outweighed by the fact you're not using any charge cycles.  However, this is a purely speculative assumption and I could find no evidence to either confirm or deny.  If this is NOT true, and the AC adapter only goes to charging the battery, then there would be no difference. Your battery would drain to 99%, then back up to 100%, then back down to 99%, etc. Those micro-charge cycles would add up at the same rate as 100% -> 0% -> 100%, and you would get no gain. In this case the negative effect of holding LiPo batteries at 100% would outweigh everything else.  Elaborating on Option 2  There are lots of good reasons why Option 2 is the way it is:   Because of known LiPo chemistry issues, holding a 100% charge for a long time causes the battery to degrade Apple specifically recommends not to leave it plugged in all the time  Draining a battery all the way to zero all the time is bad for the cell (Which is why ~10% is used) Heat is a killer, and when plugged in AND charging, you get extra heat that causes damage.   There are a couple reasons why I'm challenging Option 2:   It seems silly that the most intuitive use case for thousands of people that use their macs as primary computers is wrong. If the hypothesis in option 1 is true, you're just burning unnecessarily through your finite charge cycles. It's a pain to have to remember to keep plugging in and plugging out and be worried about whether or not your device is plugged in.