I'm trying to prove the result that $$\sum_1^\infty \frac{1}{n^2}=\pi^2/6$$ using cotangents and residue theory.  I know that $\sum f(n)=-$Sum of residues of $\pi \cot (\pi z)f(z)$ at the poles of $f(z)$  $$\sum_1^\infty \frac{1}{n^2}=\frac{1}{2}\sum_{-\infty}^\infty \frac{1}{n^2}$$.  Now the only pole is $z=0$ which is a double pole, but when I evaluate:  $$\frac{d}{dz} \pi \cot (\pi z) \frac{1}{z^2}(z^2)=\infty$$  Can somebody show me why my method doesn't work?
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?
This plant was found growing in Germany. I have never seen it before, reverse Google image search did not help either. Can you help me identify what type of plant this is?      Thanks, looks like it is something like this:    Related question: Study the following limit: $$ \lim\limits_{ (x,y,z) \to (0,0,0) } \frac{xy-z^2}{x^2+y^2+z^2} $$   I'm not sure if this way of proving that the limit does not exist is correct so I'm this is basically for a proof verification.  Approaching $(0,0,0)$ through the plane $z=0$, then the limit becomes  $$ \lim\limits_{ (x,y) \to (0,0) } \frac{xy}{x^2+y^2} $$  Now this limit does not exist, because approaching $(0,0)$ through the line $y=0$, the limit becomes  $$ \lim\limits_{ x \to 0 } \frac{0}{x^2} =0$$  But if we approach $(0,0)$ through the line $y=x$ we have that  $$ \lim\limits_{ (x,x) \to (0,0) } \frac{x^2}{2x^2}=\lim\limits_{ (x,x) \to (0,0) } \frac{1}{2} =\frac{1}{2}$$  So the limit $ \lim\limits_{ (x,y) \to (0,0) } \frac{xy}{x^2+y^2} $ doesn't exist. This means that the limit we began with doesn't exist, correct? A curious integral The following integral has been on my mind for a while $$\int_0^\infty \frac{\sin(x)}{e^x-1}\,\mathrm d x \tag{$\dagger$}$$    Let us indicate the integrand as $f(x)=\frac{\sin(x)}{e^x-1}$. The following are a couple of observations.   The integrand can be extended by continuity in $0$ since $$\lim_{x\to 0}f(x)=\lim_{x\to 0}\frac{\frac{\sin(x)}{x}}{\frac{e^x-1}{x}}=1$$ This is the original reason I started playing around with this integral. Mathematica yields the result $$\int_0^\infty \frac{\sin(x)}{e^x-1}\,\mathrm d x=\frac{\pi}{2}\textrm{Coth}(\pi)-\frac 12 \approx 1.076674047$$ which, following numerical evidence, seems correct. Complex analysis may be useful here, since the integrand is a holomorphic function on $\mathbb C$. I tried writing $\sin(z)=\frac{e^{iz}-e^{-iz}}{2i}$, but I was not able to find an appropriate integration contour to solve the problem. The $-1$ in the denominator breaks the simmetry of the expression. This made most of my substitutions useless.   Can this integral be evaluated correctly, preferrably through complex analytic methods? How do I get adequate depth of field in macro photography? I have been messing around with some Macro Photography using the techniques (and a reversal ring) described in a  by @ My main issue is that I am having a hard time getting the subject into focus. There are times when the center is blurred and the outside is in focus or the center in focus and the outside out of focus. What are ways to get as much of the subject in focus?   I am using a Nikon D5100, I have a 18-55mm kit lens but I find the 55-200mm lens easier to use for shooting. What is the value of $1^i$? What is the value of $1^i$? $\,$ I'm looking for a way to prove $$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{(-1)^m E_{2m}\pi^{2m+1}}{4^{m+1}(2m)!}$$ I know that $$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{1}{4^{2m+1}}\left(\zeta\left(2m+1,\frac14\right)-\zeta\left(2m+1,\frac34\right)\right)$$ so maybe I could simplify the above more? How do I backup my Ubuntu Phone? Bootstrap Paradox in Interstellar I am seeking suggestions on what may be going in with an iMac I am currently troubleshooting, as it’s displaying behaviour I do not recall seeing before.  Short story  This iMac refuses to boot up. It gets the Apple logo, gets past the progress bar, and then gets stuck on the grey screen for about two minutes before rebooting itself. However, what makes this odd is that I can:   Start the iMac in Target Disk Mode and use it to successfully boot up other Macs. In doing so everything seems to work perfectly fine. Successfully boot the iMac from an external drive and everything seems to work perfectly fine.   Long story  I was presented this iMac about 2 weeks ago as it refused to bootup. That is, it would commence booting up and cycle through the Apple logo and progress bar, but after the progress bar completed it would remain on the grey screen for a couple of minutes before rebooting itself and doing the same thing all over again.   To address this I undertook the following troubleshooting steps:   Resetting SMC - failed Resetting NVRAM - failed Starting up in Safe Mode - failed Starting up in OS X Recovery Mode – failed Starting up in OS X Recovery Mode via the internet – failed Undertook an extensive hardware test using Apple Hardware Test (AHT 3A213) - passed Started iMac in Target Disk Mode and retested hard drive using Scannerz on another iMac - passed Started iMac in Target Disk Mode and reformatted the hard drive and did a clean install of Mac OS X El Capitan 10.11.6 - successful Migrated data from a Time Machine backup - successful   After these steps I tested the iMac for four days with no issues in terms of booting up and using the computer. On the fifth day I needed to manually import seven Apple Mail mailboxes. I did these one by one and six of them imported within minutes.  The last mailbox was showing as being 270 GB in size. I knew this was inaccurate, but I decided to import the mailbox so I could do a mailbox rebuild to correct whatever corruption had taken place. This import took over 12hrs to do but seemed to work fine and Mail was displaying the emails.  I then tested the iMac on and off for another day to ensure it had no problems booting up. This was successful.  I then decided to initiate the mailbox rebuild, but as soon as I did the screen went black and stayed that way regardless of any keyboard shortcuts I tried using. I waited an hour, forced a reboot, and ever since the iMac has returned to its original behaviour of not booting up – it would get past the progress bar and stay on the grey screen for a while before rebooting itself and starting the whole vicious cycle again.  I have again retested the hard drive, processor, RAM and Video Ram by starting the iMac from an external hard drive and using a combination of the following packages:   Apple Hardware Test (AHT 3A213) Scannerz TechTool Pro Memtester (under the Homebrew subsystem)   And now I have a situation where I can boot the iMac from external hard drives, and also use it in target disk mode to boot up other Macs, but cannot use it to boot from its own internal drive.  At present I have isolated the internal hard drive and am running a scan to check for any malware.   In the meantime, I am wondering:   Has anyone come across a similar scenario and, if so, what was the cause? Assuming I find no malware, do I go through the reformat/clean install/migration process again, but this time not import the last mailbox and see whether the iMac works fine? Any other suggestions?   iMac specs are as follows:   iMac (27”, mid 2011) 3.1 GHz Intel Core i5 8 GB RAM AMD Radeon HD 6970M 1024 MB 1 TB 3.5” SATA HD Mac OS X El Capitan 10.11.6 How do I upgrade an iPhone 4 from iOS 7.1.2 to iOS 8 or 9 I have an iPhone 4 and I need to upgrade from 7.1.2 to iOS 8 or 9. I would like to ask how I can get iOS 8. UV mapping a straight image around a ring shaped flat mesh Is it possible to mapping a image around a ring shaped mesh? I would imagine there should be a way to curve/distort the image around the mesh. If not, how can I straighten out the mesh easily instead to fit the image? LM317 getting hot Should an LM317 (specifically a ) rated for 500ma output current really be getting hot to touch when only pulling 170ma at 5V?  Package is SOT-223, input is 9V.  Is this normal? Let $(e_n)$ (where $ e_n $ has a 1 in the $n$-th place and zeros otherwise) be unit  standard vectors of $\ell_\infty$.   Why is $(e_n)$ not a basis for $\ell_\infty$?  Thanks. You'd think that as soon as Harry met someone who was friends with his dad, he'd have asked about his family.  Certainly Lupin or Sirius could have told him something of his family, given that they were James' closest friends, and Sirius was "always welcome at the Potters".  Obviously there's nothing in the books, but did J.K. Rowling ever explain why there was no mention of any living relative other than Aunt Petunia, or even what happened to Harry's grandparents? Best books for mathematical background? Visiting Iran as a dual EU-American citizen How to use a C++ string in a structure when malloc()-ing the same structure? $\infty = -1 $ paradox When I render a page using the Django template renderer, I can pass in a dictionary variable containing various values to manipulate them in the page using {{ myVar }}.  Is there a way to access the same variable in Javascript (perhaps using the DOM, I don't know how Django makes the variables accessible)? I want to be able to lookup details using an AJAX lookup based on the values contained in the variables passed in. How can I let people join my minecraft server from diffrent versions?