Is 1-1+1-1+1-1+...=1/2 really true?

I did a bit of reading into this. This is grand series, right? And aren't both option B and C correct? As both are fine approaches to the problem. The correct answer is 1/2 as per the solution set. Okay, let's take a look. So yes, this is the famous Grundy series. It goes like this. One minus one + one minus one and so on so on. First let me just tell you guys what my answer to this question is right away is option D. None of this. And the reason for that is because this series diverges in the usual sense. Diverges means that this right here does not converge to a finite value. So it's going to be none of this. All right. But I do want to show you why and how people might interpret this series. So let's have a look. First maybe we can do the following when we have 1 - 1 + 1 - 1 and so on so on. Well 1 minus one is just zero and then if you look at 1 - 1 that's again zero and then 1 minus one it's of course zero and if you continue you're just adding infinitely many zeros. So the result is just equal to zero right? So that it looks like we have option B for the answer. But what if we do the following instead? Second way 1 - 1 + 1 - one and so on so on. Instead of putting the one and minus one together and guess what if we save the first one? It's just like when you make the first dollar, you want to keep it and then you want to borrow $1 from somebody else and then use it. So min -1 + one that gives you zero and then min -1 + one minus one the next would be a plus one I just didn't write it down that would have been a plus one so that would be zero and so on so on in this case we should end up with just one right so that suggests us option a is the answer hm H okay now we have some disagreement and usually when this kind of things happen you can see that the truth is the n partial sum does not converge to a finite value that's why this right here diverges so that would be it how do people get 1/2 you can look at it this way first zero and one maybe you want to make everybody happy so why don't we just say the answer is 1/2 which is the average of zero and one. Everybody should be happy in that sense, right? Well, maybe not. Okay, let's talk about more legitimately way to see why and how we end up with 1/2. So, the third way, you actually have two ways to look at how we can get this 1/2 right here. That's But I'm going to just do this right here. 1 - 1 + 1 minus one and so on so on. We can first look at this as a geometric series. Put in the series form. Let's use n starting at zero and it goes on to infinity. To get to the next number, we just multiply by -1. That's the common ratio we putting inside here. And we just have to raise that to the n's power because I started at n is equal to zero. Plugging zero1 to the first power we get one. Plugging one, we do end up with negative 1. Plugging two is positive one. And so on so on. And let me just remind you guys right here. Note when we have a geometry series with a common ratio r and raised to the n's power, this right here should just be the first term which plugging zero into n, r to the zero power is one over it's always one right here. and then minus the common ratio. That's a formula for adding up the geometric series. So maybe we can do that right here. The first term is one and then the common ratio is -1. So if you do that, that is 12, right? So maybe we should just use the geometric series formula to get the answer. So answer is C. Well, no. This is actually not true because the reason is because this right here because in order for us to use this formula, we have to be careful with the

detail. The detail is that this only works if the absolute value of the common ratio is less than one. Meaning under this condition this right here converges. If you don't have does not converge. That means you cannot use this formula. In our situation here r is equal to -1. If you take the absolute value of course this right here is one which is not less than one. That means we are not allowed to use this formula. So in fact C is also incorrect. A B C are all incorrect in the usual sense. But now I do want to mention the following. Assuming you are just taking a pre-calculus class or calculus class, then that's how you answer it. This right here diverges than deal. However, sometimes we do want to continue with the math. Sometimes even though we have a divergent series, we still want to give an answer to it. In that case we say the following. This is depending on if you're taking like a higher level math right. So this right here is called the cerial summation. In that case what we will do is 1 - 1 + 1 - 1 and so on. We are going to assign. So this right here we're just going to assign an answer for this series. And we will assign 1/2 to be the answer for that. Well, you might be wondering why. Why not assigning zero? Why not assigning one? We assign some value to something that we didn't have an answer to. And the truth is that this is not the first time you are seeing this kind of things because before when you saw square root of -1, how would you answer this? You will say this right here has no answer, right? You cannot have square root of a negative number. However, technically speaking, this right here has no real answer because we can answer this as I which is the imaginary unit. I is defined it to be I squared giving you -1 and I is called the principal square root of -1. So we assign something to something that we didn't have the answer to. Why did we do this though? Well, in fact once we did this we can actually continue with the math even though I is called imaginary unit but we do have real life applications with I. You can look at some differential equations, uh, things like that. So the truth is you can actually continue with this kind of series. There's a fun one for you guys. If you do 1 + 2 + 3 + 4 and so on, so on so on. In the usual sense, this right here is just going to give you infinity, right? But you can assign the value 1 /2 for the answer for this. You can look up Rammanuja summation. the SATA function. You can look up a video by member file. I will link the video in the description for you. And one more before we go. If you have taken some calculus integrals, then the integral let's say we have negative infinity to infinity of x. Technically this integral diverges. It has no answer for it. But we can say this is equal to zero. And again this is more like a higher level math. We will say this as the Koshi principal value and put a CP for this equality. That's it.