Recently, sources for mathematical infotainment, for example , have given some information on the interpretation of divergent series as real numbers, for example  $\sum_{i=0}^\infty i = -{1 \over 12}$  This equation in particular is said to have some importance in modern physics, but, being infotainment, there is not much detail beyond that.  As an IT Major, I am intrigued by the implications of this, and also the mathematical backgrounds, especially since this equality is also used in the expansion of the domain of the Riemann-Zeta function.  But how does this work? Where does this equation come from, and how can we think of it in more intuitive terms?
$\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) &gt; 1$.  Why should analytically continuing to $\zeta(-1)$ give the right answer?
I'm in trouble with this homework.  Find the sum of the series $ 1 + \frac12 + \frac13 + \frac14 + \frac16 + \frac18 + \frac19 + \frac1{12} +\cdots$, where the terms are the inverse of the positive integers whose only prime factors are 2 and 3.  Hint: write the series as a product of two geometric series.  Ok, I found some patterns in these numbers, but I can't find the two series. Dublin transit visa requirements for Indian citizens Is there any way to fix the wavy lines that appear when photographing a striped dress?   It's not just in viewing online: when it was printed at wallet size, the wavy lines showed up. Any examples of life forms on Earth that could have survived on Mars or Moon? For the command   sudo apt-get update   I get following error:  W: Failed to fetch http://ch.archive.ubuntu.com/ubuntu/dists/maverick-updates/multiverse/binary-i386/Packages.gz   407  Proxy Authentication Required (The ISA Server requires authorization to fulfill the request. Access to the Web Proxy filter is denied. )   I am running Ubuntu 10.10 installed on Windows XP using VirtualBox. For Internet connections I am using proxy server with an authentication. I tried to use gnome-network-proxy tool to set proxy settings system-wide.   After that /etc/environment has been updated by http_proxy variable with the format http://my_proxy:port/, there were no authentication data.  I checked this with Firefox. Browser asked my for login and password and everything was working fine. It was unfortunately not the case for apt-get. I have also tried to do as described . Unfortunately it does not work.  May it be somehow related to the fact that a proxy is in a Windows domain, any ideas?  My proxy name is http-proxy. Is '-' a special character here? Will Determinism be ever possible? Multiple Image in Canvas ItemsControl I have created this with cycles:  But the edge between the glowing and non glowing materials is very sharp. The material was manually applied to faces in edit mode. Is there any way I can smooth the edge between them in cycles? Javascript object members that are prototyped as arrays become shared by all class instances I've been wondering, what is the difference between seek and search? When should one be preferred over the other? I do not have enough time to bake my bread.  Can I refrigerate it or let it sit on the counter and bake it later in the day? How to make Blender use Python2.x? Do I need to change my ESTA address when visiting the US? I have an ESTA for my visit last Sept to NY and I know it is valid for 2years.  I am going to Las Vegas in May, do I have to update the address at which I will be staying? Looking around for rim brake pads, I have noticed a wide variety:   low end-ish: cheap, rubber pad with metal stem, unknown brand high end-ish: expensive, rubber pad with metal stem, known brand cool end-ish: dedicate pad sections (1 for dry, 1 for mud, 1 for rain)   Is there any good reason to go for an high end-ish or for the cool end-ish, or is it just plain old marketing tricks? Can prematurely powering down a computer cause bad sectors in a hard drive? Q1: Can forcibly powering off a PC without using the built-in shutdown/reboot functions, cause a HDD to accumulate bad sectors? Whether it be the result of BSODs, lockups, crashes, etc.   Q2: If so, are these bad sectors repairable? I know an OS typically relocate bad sectors to other areas of the drive, but once a drive accumulates too many it just dies.   I have read up and saw things about 'soft' and 'hard' bad sectors. One caused by software and the other being actual physical damage. The reason I am asking this is because I have had several HDDs die on me in the past year, far before their estimated lifespan, and one thing they all had in common was that I had forcibly powered off my PC a lot. Find the Resistance between Two Terminals Let's say R = 49 ohms, and we want to find Req between a and d, how do we do that? Do we have to find Req for the whole circuit? Does the series $\sum_{n=1}^{\infty}|x|^\sqrt n$ converge pointwise? If it then what would be the sum? Does the series $\sum_{n=1}^{\infty}|x|^\sqrt n$ converge pointwise? How can I change GNOME Shell theme? Referee says the proof is wrong, but it is not, what to do? About 3 months ago, my co-author and I submitted a mathematical paper to a journal. Yesterday, we got the editor's response and the paper has been rejected. The reason is that the referee thinks that our results are wrong or trivial.  However, we carefully read the referee's report and we conclude that the "errors" pointed out by the referee are not errors at all and that our proofs are correct. In all modesty, we think that the referee understood very poorly even the statements of our theorems.  Personally, this is the first time that such a situation occurs to me. I already have four papers published in such journal and the reports I got from the referees have been always really accurate.  What do you think is the best thing to do? Forget about and try another journal? Write an e-mail to the editor explaining why we disagree with the decision? Other ideas?  EDIT: Thank you very much for all your answers. At the end, we submitted our paper to another journal with only some slight edits we have already planned to do independently by the referee's report. Chrome and IE 11 are fine, I'm only able to reproduce this in Firefox. Using version 27.0.1 on Windows 7. Meta as well as all the other sites are affected.        I can still see the hover effect, but the image is gone. I see this in Firebug:        The image doesn't show up in the Network tab. I am able to see the image by visiting  which goes to   I've done Ctrl-F5, I've cleared the cache and I've tried Private Browsing. Possibly related to  ?