1) Form 1e (Application: Fill & Download for Free

(In the same vein as at How would you explain why 0+1-2+3-4… equals ¼ to a ninth grader?)The sense in which 1 + 2 + 3 + 4 + ... = -1/12 is this:First, consider X = 1 - 1 + 1 - 1 + .... Note that X + (X shifted over by one position) = 1 + 0 + 0 + 0 + ... = 1. Thus, in some sense, X + X = 1, and so, in some sense, X = 1/2.Now consider Y = 1 - 2 + 3 - 4 + ... . Note that Y + (Y shifted over by one position) = 1 - 1 + 1 - 1 + ... = X. Thus, in some sense, Y + Y = X, and so, in some sense, Y = X/2 = 1/4.Finally, consider Z = 1 + 2 + 3 + 4 + ... Note that Z - Y = 0 + 4 + 0 + 8 + ... = (zeros interleaved with 4 * Z). Thus, in some sense, Z - Y = 4Z, and so, in some sense, Z = -Y/3 = -1/12.In contexts where the above reasoning is applicable to what one wants to call summation, we have that 1 + 2 + 3 + 4 + ... = -1/12. In other contexts, we don't.That's it. It's that simple. Everything else I'm going to say is just to comfort those who are uncomfortable with the game we've just played.Note that I've said "in some sense" several times in the above argument. That's because, while we all know how to add and subtract a finite collection of numbers in the ordinary way, when it comes to adding and subtracting an infinite series of numbers, there are many different ways of interpreting what this should mean. Just knowing how to add finitely many numbers doesn't automatically tell us what it means to add a whole infinite series of them. And when it comes to summation of infinite series, it turns out there's not just one nice notion of "summation"; there are many different ones, which are nice for different purposes.One such notion is "Keep adding things up, one by one, starting from the front, and see if the results get closer and closer to some particular value; if so, that value is the sum". On that account of what summation means, you clearly won't get any finite answer for 1 + 2 + 3 + 4 + ...; since the terms never get any smaller, the partial sums will never settle down to a finite value (and certainly not a negative one like -1/12!). They instead, in a natural sense, should be understood as summing to positive infinity.And there's nothing wrong with this! You are not wrong to feel that 1 + 2 + 3 + 4 + ... is positive and infinite, and math does not deny this; there absolutely is an account of summation corresponding to this intuition.It's just not the only account of summation worth thinking about.We could instead consider other notions of "summation", including ones designed precisely so that arguments like the one we made at the beginning (which are very natural arguments to make!) counted as legitimate ways to reason about such "summation". And then, by definition, we will have that 1 + 2 + 3 + 4 + ... = -1/12, on such accounts of "summation".(In doing so, we will lose certain familiar properties such as "A sum of positive terms is always positive". But this is how generalizations work; generalizations very often lose familiar properties. Even the textbook, limit-based account of infinite summation loses familiar properties like "The order of summation doesn't matter". Even finitary summation of integers loses the familiar property "If a sum is zero, so are all the summands" from basic counting. But there is a web of resemblances to more familiar kinds of summation which can justify, in certain moods, thinking of each of these generalizations as a form of summation itself.)If you insist that "Keep adding things up and see if the results get closer and closer to some particular value" is the only account of summation you're interested in, you'll object to the argument we gave at the beginning, saying "You're not allowed to do that kind of shifting over and adding to itself reasoning all willy-nilly; look at what nonsense it produces!".But it can be made sense of, and is even fruitful to make sense of, in certain contexts in mathematics, and there is no need to blind ourselves to this insight.Again, that's it. It's that simple. Everything else I'm going to say is just to comfort those who are still uncomfortable. For those who want a more systematic, formal account of series summation of a sort which validates the above manipulation, read on:We can look at it this way: We can try to assign values to a non-absolutely convergent series by bringing its terms in at less than full strength, producing an absolutely convergent series, and then increasing the terms' strengths towards full strength in the limit, observing what happens to the sum in the limit as well.This is the idea behind the traditional account of series summation, mind you: at time T, we bring in all the terms of index < T at 100% strength and all other terms at 0% strength. This gives us our partial sums, and as T goes to infinity, each term's strength goes to 100%, so we can consider the partial sums as approximating the overall sum.But we don't have to be so discrete as to only use 100% strength and 0% strength. We can try bringing in terms more gradually. For example, rather than having strengths discretely decay from 100% to 0% at some cut-off point, we can instead have the strengths decay exponentially in the index. (So at one moment, we may have the first term at 100% strength, the next term at 50% strength, the next term at 25% strength, etc.). Then we consider what happens as the rate of exponential decay slows, approaching no decay at all.In symbols, this means we assign to a series [math]a_0 + a_1 + a_2...[/math] the limit, as [math]b[/math] approaches [math]1[/math][math][/math] from below, of [math]a_0b^0 + a_1b^1 + a_2b^2 + ...[/math]. Put another way, the limit, as [math]h[/math] goes to [math]0[/math] from above, of [math]a_0e^{-0h}+ a_1e^{-1h} + a_2e^{-2h} + ...[/math], where [math]e[/math] is any fixed base you like. (Let's take [math]e[/math] to be the base of the natural logarithm for convenience, and call this function of [math]h[/math] the characteristic function of the series).Again, this is not so different than the traditional account of series summation; we're just using exponential decay rather than sharp cutoff in our dampened approximations to the full series. (Actually, for the results we're interested in, it's really just the smoothness of the decay that's of interest. We could use other forms of smooth decay as well, and get the same results, but exponential decay is so convenient, I won't bother discussing in any further generality right now)Now we've turned the question of determining the value of a series summation into the question of determining the limiting behavior of some function at 0.Well, it's easy to determine limiting behavior at 0. Just write out a Taylor series centered at 0, and drop all the terms of positive degree, leaving only the term of degree 0. Boom, you've got the value of the function at 0.Except... suppose the Taylor series has a few terms of negative degree as well. (As in, say, [math]5h^{-1} + 3 + 4h^2[/math]). Then the behavior at 0 isn't given by the degree 0 term; rather, the behavior at 0 is to blow up to infinity!And, indeed, we'll find that this is precisely what happens when we look at the characteristic function of a series like 0 + 1 + 2 + 3 + ...; we get that [math]f(h) = 0e^{-0h} + 1e^{-1h} + 2e^{-2h} + 3e^{-3h} + ...[/math][math] = e^{-h}/(1 - e^{-h})^2 [/math][math]= h^{-2} - 1/12 + h^2/240 - h^4/6048 + ...[/math].Note that there is a negative degree term there. So in a very familiar sense, we can say that the behavior of this series is to blow up to infinity.However, since any time a series DOES converge in the ordinary sense, the value it converges to is the degree 0 term of this characteristic function, it is very tempting and fruitful to think of the degree 0 term as the sum even when there are those pesky negative degree terms.And in this more general sense, we see that the value of 0 + 1 + 2 + 3 + ... is that degree 0 term of f(h): -1/12. [In fact, we can understand the argument at the beginning of this post as outlining a rigorous calculation of this degree 0 term. (See https://www.quora.com/Whats-the-intuition-behind-the-equation-1+2+3+-cdots-tfrac-1-12/answer/David-Joyce-11/comment/3444455 to see this spelt out)]Now, you can propose other manipulations to produce other answers for this series in other ways, but this is one particular systematic account of summation which leads to this value alone and no other. [That is, for the series whose nth term is n. I should warn that, in the presence of negative degree terms in the characteristic function, this method is sensitive to index-shifting, so we would get different results if, for example, we considered 1, 2, 3, ... to be not the 1st, 2nd, 3rd, ..., terms, but rather the 0th, 1st, 2nd, ..., terms, respectively.]Why should you care about this particular account of summation? Well, you don't have to; I can't force you to care about anything. But it's fairly natural and comes up with some significance in mathematics. It is, in a certain formal sense, precisely the account of summation which allows one to interpret the sum [math]1^n + 2^n + 3^n + ...[/math] for general complex [math]n[/math], yielding the Riemann zeta function (of great significance in number theory, and whose behavior (specifically, the Riemann hypothesis concerning its zeros) is generally considered one of the most important open problems in mathematics). So, you know, there's reason for some people to care about it, even if you don't.

Do read the whole answer.I have curated some questions..which will help you understand many things about IRNSS, like:What is IRNSS?Why do we need IRNSS?Will IRNSS be more accurate than GPS?Budget, Applications, Time frame of IRNS & etc.Not for ISRO, I guess. Kudos to ISRO. :)What is IRNSS?In simple terms, IRNSS is our local GPS.The Indian Regional Navigation Satellite System or IRNSS is an Indian developed Navigation Satellite System that will be used to provide accurate real-time positioning and timing services over India region extending to 1500 km around India.It is named as NAVIC (Navigation with Indian Constellation) (the word navic (नाविक) meaning seafarer in Sanskrit).For our own GPS, we need 7 satellites, 3 in geostationary orbit and 4 in geosynchronous orbit.From ground, the three geostationary satellites will appear at a fixed point in the sky. However, the four geosynchronous satellites moving in inclined orbits in pairs will appear to move in the figure of '8' when 'seen' from ground.Fun Facts:As of now only US , RUSSIA & China have its own Navigation System. Indian will be 4th in the list [1].European Union is also working on its own Navigation System, but that will only be functional by 2020[2] .It will cover India region extending to 1500 km around India.Indian Government is also planning to help SAARC countries with Navigation Services & much more.[3][4]Why do we need IRNSS?During Kargil War, Indian army was using GPS to locate enemy, But USA refused to help Indian Army that time. So, Indian Govt. decided to built it's own indigenous positioning System[5].Our own system will also helps us in many ways:We no longer depend on US/Foreign GPS networks in terms of navigation on critical situation (cold war like period).US GPS system has dual operation , unencrypted GPS for public use and encrypted GPS for armed forces that is highly accurate when compared to civilian GPS. Only US has access to it. Now with our indigenous IRNSS , we have our own high accuracy encrypted GPS for our armed forces that covers the Indian subcontinent area.GPS guided arms like missiles could now be manufactured by Indian armed forces. Earlier it couldn't be made as civilian GPS couldn't be used for such purposes.Already existing GPS/GLONASS constellation of satellites are polar satellites that keep orbiting the entire world that reduces the number of satellites overhead. But with total of 7 GEO and GSO satellites it could provide better visibility of data to Indian mainland.IRNSS will totally be in control of Indian government.Will IRNSS be more accurate than GPS?The system is intended to provide an absolute position accuracy of better than 10 meters throughout Indian Landmass.And better than 20 meters in the Indian Ocean as well as a region extending approximately 1,500 km around India. [6]How many Satellites does it consist of?IRNSS consist of 7 satellites.IRNSS-1AIt was launched on-board PSLV-C22 on 1 July 2013 from the Satish Dhawan Space Centre at Sriharikota.IRNSS-1BIt was successfully placed in its orbit through PSLV-C24 rocket on 4 April 2014.IRNSS-1CThe satellite was successfully launched using India's PSLV-C26 from theSatish Dhawan Space Centre at Sriharikota on 16 October 2014.IRNSS-1DIt was successfully launched using India's PSLV-C27 on 28 March 2015.IRNSS-1EIt was successfully launched on January 20, 2016 using India's PSLV-C31.IRNSS-1FThis was launched on 10 March 2016 using India's PSLV-C32.IRNSS-1GIt was successfully launched 28 April 2016using India's PSLV-C33.How much was the budget for IRNSS?The total cost of the project is expected to be ₹1420 crore (US$211 million), with the cost of the ground segment being ₹300 crore (US$45 million).Each satellites costing ₹150 crore (US$22 million) and the PSLV-XL version rocket costs around ₹130 crore (US$19 million) .The seven rockets would involve an outlay of around ₹910 crore (US$135 million).[7][8][9]How much time does it took to launch the entire system (all 7 satellites)?The very first satellite (IRNSS-1A) was launched on 1st July 2013 & the last one (IRNSS-1G) was launched on 28th April 2016.So, it took approximately 3 Years to launch & successfully place the entire system.What will be the applications of IRNSS?Since, IRNSS is an Navigation System, so it can used anywhere where you need to know the real-time position of anything. Some standards uses can be:Terrestrial, Aerial and Marine NavigationDisaster ManagementVehicle tracking and Fleet managementIntegration with mobile phonesPrecise TimingMapping and Geodetic data captureTerrestrial navigation aid for hikers and travelersVisual and voice navigation for driversSome Technical Details:Three of the satellites are located in geostationary orbit (GEO) at 32.5° East, 83° East, and 131.5° Eastlangitude.The other four are inclined geosynchronous orbit (GSO). Two of the GSOs cross the equator at 55° East and two at 111.75° East.The four GSO satellites will appear to be moving in the form of an "8" .NAVIC signals will consist of a Standard Positioning Service and a Precision Service. Both will be carried on L5 (1176.45 MHz) and S band (2492.028 MHz).The navigation signals would be transmitted in the S-band frequency (2–4 GHz) and broadcast through a phased array antenna to maintain required coverage and signal strength.The satellites would weigh approximately 1,330 kg and their solar panels generate 1,400 watts.Do let me know if some information provided here is wrong/incorrect or if something more can be added to the answer by adding comments.Thanks.Pradeep Singh - www.pradeepsingh.xyzSource:Indian Regional Navigation Satellite SystemJayagopal Kannan's answer to What will be the benefits for Indians for having their own version of GPS?Images: Google ImagesFootnotes[1] Satellite navigation[2] http://spacenews.com/civil/110120-galileo-assessment-punches.html[3] ISRO looking to extend services to SAARC countries | Latest Tech News, Video & Photo Reviews at BGR India[4] India’s positioning satellite to serve South Asia[5] How Kargil spurred India to design own GPS - Times of India[6] http://www.unoosa.org/pdf/icg/2008/expert/2-3.pdf[7] Isro to launch 5th navigation satellite on Jan 20, first in 2016[8] Isro to launch 5th navigation satellite on Jan 20, first in 2016[9] India's first dedicated navigation satellite placed in orbit

Can the cantrip mending affix any surface to any other surface in D&D 5E?No, because that’s not what the cantrip does, Dex Jackson.Source: Roll20It’s a cantrip. These spells are the magical equivalent of a musician practicing scales or arpeggios: it isn’t meant to do anything all that powerful.What you appear to want is a magic item that date back to the 1e days.Sovereign GlueThis viscous, milky-white substance can form a permanent adhesive bond between any two Objects. It must be stored in a jar or flask that has been coated inside with Oil of Slipperiness. When found, a container contains 1d6 + 1 ounces.One ounce of the glue can cover a 1-foot square surface. The glue takes 1 minute to set. Once it has done so, the bond it creates can be broken only by the application of Universal Solvent or Oil of Etherealness, or with a wish spell.Source: Roll20And please note that sovereign glue is a legendary-rarity item, meaning it’s intended for characters of 17+ level.With this answer, I am joining a chorus of Quorans that includes Bram Haenen, User-13279148522749471787, Fred Huber, and Christopher Crespo. They are all exactly correct, too.