the most famous Ramanujan sum 1+2+3+...=-1/12

okay in this video let's talk about the sum of all the past the phone numbers namely 1 plus 2 plus 2 plus 4 plus dot dot this right here it's a really famous debate because we have two popular ways to answer this the first answer is that we can just say this right here diverges and the second way is this right here it's equal to negative 1 over 12 and perhaps you guys can play numberphile for the negative 1 over 12 if I have to answer this question no you will be depending on which class I'm teaching if I'm just teaching like algebra or a calculus class I will just say 1 plus 1 plus 2 Plus 3 plus 4 plus dot a dot this right here the sum is infinity therefore this diverges done deal however if you want to do more just like a typical mathematician then there will be ways that you can learn that this right here it's actually equal to negative 1 over 12 if you were on to learn more and you guys can check out 3 blue one prong and numberphile video they have their ways to talk about why the negative 1 over 12 but here let me show you how can actually use an integral to make sense of this right here and we will let me write down the formula first we will be using the new job summation and what this is is that this is just a way to assign value to a type virgin like this this right here diverges therefore we can use this little formula to keep it for you to it right and let me just write on a formula for you guys first of all you have to turn your function f of zero and then you divide it by two for that and then the next part is you add the sum as n goes from 1 to infinity f of N and this right here it's that simple to equal I will put down equal with an R on the top because this is not the regular stump in the usual sense you will be thanks to the ramanuja or as I'll be putting that equal with our on the top well this is really cool because first of all I will to use the eye again as I told you we will have the integral and this right here is improper integral going from 0 to infinity and not only we have I your still it wrong okay we'll have more things coming up but I mean right down the numerator we will have F of I T and then minus F of negative I T and then divided by well if you have I don't forget to in fact e to come out to help us out here if the e right and we also have a regular number two and of course for you see e and then I how can we not have the pie right and here's the pie and then we have the T and then we have the minus one and then that's it for this integral so now I will show you guys how we can use this right here to make sense of negative one over twelve to do so we have to come with a function of course relating this to that I can just simply pick f of X to be X star let me just write this down right here let f of X equal x and when you push the rope right here f of zero just 0 over 2 so you have 0 over 2 and then you add this is the sum as n goes from 1 to infinity and you see when f of X is simply just X F of n you put n into X which is just n so just get the sum of and when n goes from 1 to infinity namely you pretty much have this right here this is equal to with an R on the top all right I times the integral from 0 to infinity f of I T you just put I team to this X so just get I T and then minus 4 F of negative I we will just have to put in negative 2 here so that's of course just negative I'd like this pretty nice huh and then over this which is e to the 2 pi t minus 1 and now we just have to take care of this integral you see on the top we have ite plus I T so that's 2i t like this and 2i we can multiply with the I write just a constant multiple so you see items I'd negative 1 so I will write down negative 2 all the way in the front and notice I didn't put down the are right here because this part it's equal to this part in the regular sense I don't need these are anyway negative 2 times the integral from 0 to infinity on the top now I have the T over on the bottom we have e to the 2 pi t minus 1 and I just have to figure out this integral now right well to do so I will just need a little you stop right so I'll just put this down right here and what do have to 6 in your head so I will take you equal to PI T in the front we have the negative 2 and then notice this integral is going from T equals 0 to infinity when T is 0 put it here you will be 0 as well so in the new world u goes from 0 and you put infinity to here 2 pi times infinity is still infinity so you will be going to infinity right here okay T divide both sides by 2 pi so we get u over 2 pi over e to the 2 pi T and that's exactly that you so we have e to the u minus 1 and then for PT you can just you know divide of 2 pi on both sides and differentiate both sides we get D u over 2 pi like this and now we can actually cancel things out just a - this and that - and this - maybe all right let's see what else can we do this - and the PI D R in the denominator so is this PI so we are PI times pi in the denominator we have PI squared so altogether we have naked here right here and then 1 over 2 pi squared okay and then for the rest we have the integral from 0 to infinity everything is in that you were already and on the top I just have to you over here we have e to the u minus 1 and this is with tu right here now I'll write this down we have negative 1 over 2 pi square for this integral in fact I have a video on this already this is Cody balls the boss boss integral ok so you're going to check that out this it's equal to a famous number PI square over 6 okay I didn't make this up check out the video this is equal to that and from here yes you can see that the PI square cancel each other precisely and then just do a little multiplications you end up with negative 1 over 12 aha just like that so in other word this right here thanks to Ramanuja we indulged negative 1 over 12 so the conclusion based on this is that I will actually write down 1 plus 2 Plus 3 plus 4 plus dot dot dot put R on the top okay and then you have a negative 1 over 12 this right here is perhaps the conclusion that you want to say so if you want to say this is negative 1 over 12 you can say because of ramanuja so this right here once again it's just a way to assign values to divergent series yes you have 1 plus 2 Plus 3 plus 4 plus 2 dot from here you can actually do a few more things what will be the value for 1 square plus 2 square plus 3 square plus 4 square and so on that would be zero how do you do it just this okay check that out just do it on your home or so and before we end this video yes you'll see that if you have Ethan powers here the thumb it's actually syrup just tying the three blue one prong video yes that's really cool and in fact I promote to make because in his video he mentioned it the remark SATA function and here I also have that they'd have function for you guys if you look at this integral in fact this right here it's equal to gamma so it's not just the zeta function we also have the gamma function right here its gamma of two times haha here is the zeta function theta of two right here gamma 2 is one set up two years PI squared over 6 and how do we know this mishegas check out my other video for that alright and one last thing this right here it's not the first time that we have seen that we are saying some value to something that mean that makes sense in the first place in fact if you gets two more examples that you will be able to relate to this and that the first example is that imagine if you have to calculate square root of negative 9 well what do you do if you haven't seen imaginary numbers or complex numbers how would you answer the question well in this case we'll just answer that this right here has no real value because as we all know we can have negatives of the square roots however once you learn about complex numbers you can say this right here it's equal to 3 I and just the principal value okay some people may argue this is plus minus 3i up to you right so it depends on which class that you are in which map you have seen right and another example to give you in fact a video and this is that the integral from negative 1 to 1 1 over X DX for this right here we also have two ways to answer it the first way is that we can simply say this improper integral diverges and you guys can check out my other video for this right yes well and that's good for Cal to students okay but once you get into like some upper division left for math classes once you learn about what they call the Cauchy principle value you can say this right here is equal to zero and once again this is the Cauchy principle value so as you can see we have done this twice before and this time we may be able to add this to this we have two ways to answer the question depending on which class that you are in right so hopefully gets all like this video and leave a comment down below and let me know if you guys have any questions and be sure you guys watch my next video because I'll show you how we can make 1 + 2 + 3 + 4 + start making that equal to negative something else anyway that's always that's it

Channel: blackpenredpen

Views: 318,237

Keywords: blackpenredpen, math for fun, calculus tutorial, sum of all natural numbers, -1/12, Ramanujan Summation, bose integral, riemann zeta function, analytic continuation, 1+2+3+...

Id: U_g_OZLt4OU

Length: 11min 40sec (700 seconds)

Published: Fri Dec 28 2018