(I was requested to edit the question to explain why it is different that a proposed duplicate question. This seems counterproductive to do here, inside the question it self, but that is what I have been asked by the site and moderators. There is no way for me to vote against their votes. So, here I go: Please stop voting this as a duplicate so quickly, which will eventually lead to this question being closed off. Yes, the other question linked to asks the same math, but any newcomer to the problem who was exposed to it via physics, as I was, will prefer this question instead of the one that is purely mathematically. I beg the moderators to not be pedantic on this one. This question spills into physics, which is why I did the cross post to the physics forum as well.)  How does the sum of the series “1 + 2 + 3 + 4 + 5 + 6…” to infinity = “-1/12”, in the context of physics?  I heard Lawrence Krauss say this once during a debate with Hamza Tzortzis (). I found a transcript of another debate between Krauss and William Lane Craig which has the same sum. Here is the paragraph in full:     Let’s go to some of the things Dr. Craig talked about. In fact, the   existence of infinity, which he talked about which is   self-contradictory, is not self-contradictory at all. Mathematicians   know precisely how to deal with infinity; so do physicists. We rely on   infinities. In fact, there’s a field of mathematics called “Complex   Variables” which is the basis of much of modern physics, from   electro-magnetism to quantum mechanics and beyond, where in fact we   learn to deal with infinity; without the infinities we couldn’t do the   physics. We know how to sum infinite series because we can do complex   analysis. Mathematicians have taught us how. It’s strange and very   unappetizing, and in fact you can sum things that look ridiculous. For   example, if you sum the series, “1 + 2 + 3 + 4 + 5 + 6…” to infinity,   what’s the answer? “-1/12.” You don’t like it? Too bad! The   mathematics is consistent if we assign that. The world is the way it   is whether we like it or not.   -- Lawrence Krauss, debating William Lane Craig, March 30, 2011  Source:   CROSS POST: I'm not sure if I should post this in mathematics or physics, so I posted it in both. Cross post:   EDIT: I did not mean to begin a debate on why Krauss said this. I only wished to understand this interesting math. He was likely trying to showcase Craig's lack of understanding of mathematics or logic or physics or something. Whatever his purpose can be determined from the context of the full script that I linked to above. Anyone who is interested, please do. Please do not judge him out of context. Since I have watched one of these debates, I understand the context and do not hold the lack of a full breakdown as being ignorant. Keep in mind the debate I heard this in was different from the debate above.
$\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) &gt; 1$.  Why should analytically continuing to $\zeta(-1)$ give the right answer?
What is the probability that a person is the second person to roll a 6 for the first time? Suppose $A$, $B$, and $C$ roll a die in turns. What is the probability that $A$ is the second person to roll a $6$ for the first time? List manipulation: position & max value combination Is starting a group with 5 newbies (DM included) possible? Some friends and I want to start playing D&amp;D (5e), but none of us, DM included (not me) have played before to any great extent. The DM and I have some cursory experience with 4e, but have never been able to play to any great extent. Is this advisable, or should we play with existing groups first? We live in a relatively remote area, making it hard to find experienced players. Bluetooth indicator disappears on turning off bluetooth Everytime I turn off my bluetooth, the indicator icon in the system tray disappears, as opposed to just greying out as expected. Next time I need to go to the settings select bluetooth to turn it on, which makes the indicator to reappear.  Is there anyway I can make the indicator icon not disappear but stay greyed out when not in use? If $p$ is prime, then $(x+y)^p=x^p+y^p$ holds in any field of characteristic $p$. However all the proofs I have seen use induction and some relatively nasty algebra despite how fundamental this fact seems.  What is the nicest, "highest level proof" you know? It would be nice to be able to put emphasis on specific bits of a fragment or of a quote (bold can help for a quote but I find that it makes things less readable sometimes). What do you think? Update: On , you can highlight particular lines by prefixing each line with @@. See also: Geometric random variables $X_1:G(p_1)$ $X_2:G(p_2)$ $X_3:G(p_3)$ are independent, prove the following : I have a Toshiba z830 with ubuntu 12.04. I could use without problems the DSL connection (I have no router, only a modem) at home until a couple of days ago:   It stopped to work when while connected with the DSL connection via CiscoVPN (to my workplace), I had to leave the laptop unattended for a while and I guess the laptop tried to go to suspend. When I came back there were many messages on a black screen (sorry, I cannot attach an image because of my low reputation) and the laptop was not responding. I had to reset the it.   After rebooting it, the internet connection didn't work as did before (no browsing, no ssh, no skype, etc) although it says it is connected. I only can browse, etc when I connect to CiscoVPN to the same place where I was connected when it crashed.  When I am physically where I connected via VPN (at work) I can use the wireless, which I am not able to use now anymore since the incident described above.  Some additional information:  martillu@ubuntu:~$ cat /etc/resolv.conf   domain km.icrr.u-tokyo.ac.jp   nameserver 10.240.12.134   nameserver 10.240.12.135    martillu@ubuntu:~$ route -n   Kernel IP routing table   Destination Gateway Genmask Flags Metric Ref Use Iface   0.0.0.0 61.127.116.199 0.0.0.0 UG 0 0 0 ppp0   61.127.116.199 0.0.0.0 255.255.255.255 UH 0 0 0 ppp0   169.254.0.0 0.0.0.0 255.255.0.0 U 1000 0 0 ppp0    martillu@ubuntu:~$ ifconfig   eth0 Link encap:Ethernet HWaddr e8:e0:b7:2f:bc:5a   inet6 addr: fe80::eae0:b7ff:fe2f:bc5a/64 Scope:Link   UP BROADCAST RUNNING MULTICAST MTU:1500 Metric:1   RX packets:17620 errors:0 dropped:1 overruns:0 frame:0   TX packets:13168 errors:0 dropped:0 overruns:0 carrier:0   collisions:0 txqueuelen:1000   RX bytes:19452232 (19.4 MB) TX bytes:2568218 (2.5 MB)   Interrupt:20 Memory:c0700000-c0720000    lo Link encap:Local Loopback   inet addr:127.0.0.1 Mask:255.0.0.0   inet6 addr: ::1/128 Scope:Host   UP LOOPBACK RUNNING MTU:16436 Metric:1   RX packets:134 errors:0 dropped:0 overruns:0 frame:0   TX packets:134 errors:0 dropped:0 overruns:0 carrier:0   collisions:0 txqueuelen:0   RX bytes:36316 (36.3 KB) TX bytes:36316 (36.3 KB)    ppp0 Link encapoint-to-Point Protocol   inet addr:219.167.252.226 P-t-P:61.127.116.199 Mask:255.255.255.255   UP POINTOPOINT RUNNING NOARP MULTICAST MTU:1454 Metric:1   RX packets:17608 errors:0 dropped:0 overruns:0 frame:0   TX packets:13129 errors:0 dropped:0 overruns:0 carrier:0   collisions:0 txqueuelen:3   RX bytes:18993590 (18.9 MB) TX bytes:2221195 (2.2 MB)    wlan0 Link encap:Ethernet HWaddr 9c:b7:0d:d9:21:f3   UP BROADCAST MULTICAST MTU:1500 Metric:1   RX packets:0 errors:0 dropped:0 overruns:0 frame:0   TX packets:0 errors:0 dropped:0 overruns:0 carrier:0   collisions:0 txqueuelen:1000   RX bytes:0 (0.0 B) TX bytes:0 (0.0 B)    I think that: domain km.icrr.u-tokyo.ac.jp in the resolv.conf may be suspicious, since if I am not connected via CiscoVPN I do not know why it should appear "km.icrr.u-tokyo.ac.jp" (this is where I connect via VPN). When I'm on my mod site I have my diamond dropdown:    But on Meta.SE and other sites, I have the StackBurger:    This doesn't show on my mod site, so I can't switch between sites easily. Can you transit through the Schengen Area without a visa when traveling on separate tickets? I am planning to travel from US to Albania, and when I search on flight ticket websites such that US as origin and Tirana (TIA) as destination, then it costs me $1050 cheapest. However, it costs me $850 when I buy two separate round-trips through Barcelona (US to Barcelona round-trip, Barcelona to Tirana round-trip). I just checked Barcelona, but there might be other transfer airports in Schengen area which costs me even less. I don't need a visa for Albania but I need a visa to go into a Schengen country, so my question is 3-fold:   How can I learn which Europe airports have the international transfer area, so that I don't need a Schengen visa to transfer through the airport? I guess the airlines would take care of that when you buy the tickets together (I already have such an experience multiple times), but given that I will buy them separately, I need to have that information beforehand. The two separate flights I am planning to buy are through different airlines, and in my experience, the airlines check whether the traveler has a visa for the destination country before they let you into the airplane. So, would it be easy to convince the US to Barcelona trip airline that Barcelona is not my final destination, but I have a separate flight from Barcelona to Albania? Are there any Europe airports that have "international baggage claim" areas which are before the passport checkpoint? I mean, if I buy separate flights, do I have any chance to check-in a baggage for the US to Barcelona trip? I have a file with the lines like below.  title1:A1 title2:A2 title3:A3 title4:A4 title5:A5  title1:B1 title2:B2 title3:B3 title4:B4 title5:B5  title1:C1 title2:C2 title3:C3 title4:C4 title5:C5  title1:D1 title2:D2 title3:D3 title4:D4 title5:D5  How can I achieve this?   title1    title2     title3    title4 A1         A2         A3         A4 B1         B2         B3         B4 C1         C2         C3         C4 D1         D2         D3         D4 How to Toggle Selected Objects Visibility with H shortcut as in Maya I want to be able to toggle the visibility of selected objects with a shortcut (H) the way you can do in Maya and XSI, shown in the GIF. I'm guessing scrippting is needed. I've tried to do it my self but I'm new to Python and Blender and as soon as the objects are hidden they seem to no longer be in a selection state/list. Eg if I manually select the hidden object/s in the Outliner and try to print the selected object to the console, none seem to be selected. Is this possible to do with the current state of Blender 2.8? Energy conservation in General Relativity I understand that energy conservation is not a rule in general relativity, but I'd like to know under what circumstances it can still be possible. In other words, when is it possible to associate a potential energy to the gravitational field, so that the energy is constant in the evolution of the system?  Here are some examples, is there a convenient way to define energy in these scenarios?   Just a system of gravitational waves. A point mass moving in a static (but otherwise arbitrary) space-time. Equivalent (if I'm not mistaken) to a test mass moving in the field of a second much larger mass, the larger mass wouldn't move. Two rotating bodies of similar mass.   Overall, I'm trying to understand what keeps us from associating a potential energy to the metric. When we break the time translation symmetry of a system by introducing an electromagnetic field, we can still conserve energy by defining an electromagnetic potential energy. Why can't we do the same when we break TT symmetry by making space-time curved? I have read about copying files with terminal but these examples will help me a lot. So here is what I want to do:  Examples:   I have a file in /home/levan/kdenlive untitelds.mpg and I want to copy this file to /media/sda3/SkyDrive and do not want to delete any thing in SkyDrive directory. I have a file in /media/sda3/SkyDrive untitelds.mpg and I want to copy this file to /home/levan/kdenlive  and do not want to delete any thing in kdenlive directory  I want to copy a folder from home directory to sda3 and do not want to delete any thing on sda3 directory and opposite I want to cut a folder/file and copy to other place without deleting files in that directory I cut it into. Let $X$ be a beta distributed variable with parameters $a$, $b$. Let $Y$ be a beta distributed variable with parameters $c$, $d$. Let $Z = \max(X, Y)$.  Does anybody know of a fast way to compute the mean and variance of $Z$ given $(a, b, c, d)$?  $X$, $Y$ don't necessarily need to have beta distributions, but they should be similar -- in $[0, 1]$.  What I really want to find is a way to quickly approximate the distribution of the sample max of a bunch of random variables. Let $Z = \max(\{X_1, X_2,\dots , X_n\})$. If I know parameters (not necessarily beta-distribution) for $X_1, X_2,\dots, X_n$, can I quickly compute a set of parameters (same distribution as $X_i$) to form a distribution that closely approximates $Z$?    EDIT: To clarify, the purpose of this has to do with programming AI for board games. If we are evaluating a position in a chess game and we determine one move leads to a 60% chance of winning while all others have a 20% chance of winning, then we value the position as a 60% win. This is the minimax algorithm... However, what I'm curious about is whether this can be improved upon. In the simple case, we approximate: $$ \mu_c = \max(\{\mu_1, \mu_2,\dots, \mu_n\}) $$ where $\mu_c$ is the expectation from the current position and $\mu_i$ is the expectation from each of the sub-trees.  It seems like we should be able to do better than this. Can we find a function f that replaces max and closely approximates parameters for the distribution of the sample maximum? $$ (\mu_c, \sigma_c) = f\left(\{(\mu_1,\sigma_1), (\mu_2,\sigma_2), \dots, (\mu_n, \sigma_n)\}\right) $$ In this case, we still want $\mu_i$ to be the expectation from each of the nodes. However, $\sigma_i$ does not necessarily need to be variance/standard deviation; just a parameter that represents uncertainty.  Just as: $$ \max(\{a, b, c\}) = \max\left(\{\max(\{a, b\}), c\}\right) $$ We should expect: $$ f\left(\{a, b, c\}\right) \approx f\left(\{f(\{a, b\}), c\}\right) $$ So $f$ does not necessarily need to take parameters for more than two nodes.  One thing, however, is that $f$ should be relatively fast to compute. If the integrals must be calculated, I think doing double exponential integration might work well, but I'm not sure -- How can I search an Excel spreadsheet in SharePoint? Looking to be able to search an Excel spreadsheet from the SharePoint mobile app.  I have a spreadsheet with Postal Codes/Zip Codes with corresponding information. I would like to be able to search as you would in Excel.  When I upload a test spreadsheet, it opens as like a PDF. User has "Read" permissions for the document library the spreadsheet is located in.  Thanks in advance, To show for following sequence $\lim_{n \to \infty} a_n = 0$ where $a_n$ = $1.3.5 ... (2n-1)\over 2.4.6...(2n)$ How can I show   $\lim_{n \to \infty} a_n = 0$  $a_n = {1.3.5 ... (2n-1)\over 2.4.6...(2n)}$  I have shown that $a_n$ is monotonically decreasing. I thought to shown sequence is bounded from below then it automatically would converge and hence my question will be solve. But I'm unable to show its boundedness... Or there maybe another method to prove this. Thanks When do I use "I" instead of "me?" From some comments in the answers for  (now deleted, 10k only), there's confusion around the usage of I vs. me:  While the sentence, "the other attendees are myself and Steve," is agreed to be incorrect, there's confusion about whether the correct form is "the other attendees are me and Steve" or "the other attendees are Steve and I."  (I've always used the heuristic of removing the other people from the sentence, so I always thought "the other attendee is me" would be correct, instead of "the other attendee is I." Is this true, or am I using a flawed heuristic?) Discourage questions being marked as answered within an hour or so of being posted What is the significance of momentum? I just want to get an idea what  is. I know the mathematical meaning that momentum is $mv$ where $v$ is velocity. But I don't know its significance. Like I know that Acceleration is how velocity is changing with respect to time, I wanted to get feel of what momentum really is?