math brain fucking

[Hamilcar]

I don't know if some of you remember your maths ...
Here's a brain fuck: 
S = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1.... to infinity .
this sum equals 0 ( if you sum from the beginning) , or 1 ( if you start summing after the first 1 ) , or ... 1/2
because S = 1 - S ... ie 2S = 1 , ie S = 1/2 .
That's Grandi's serie...


 

[lmfao]

@Hamilcar Grandi's series is divergent. As you increase the number of terms of the series you are not getting closer and closer to a particular number, so the fact that the sum=0.5 shouldn't be a brain fuck because 0.5 is arrived at from choosing an arbitrary method of summation. 1+1-1+1-1+1... To  infinity has no real solution but something like 1/2 + 1/4 +1/8 +1/16 + 1/32 +.... To infinity has a far more real solution. Because infinity isn't an actual number (and hence "limits" as a concept are defined) , "addition"is defined as acting different for a convergent series than it is for a divergent series. The way addition is defined for a convergent series is what everyone is used to, you could say that people just decided to invent a different type of addition for summing a divergent series. Here's how 1/2 was the answer for that series you mentioned. 

Let's consider a new sequence for  which the nth term of the new sequence tells you the mean of the first "n" terms of the Grandi's series. E.g., (1-1+1)/3 = 2/3 and (1-1+1-1)=2/4. We are going to redefine addition in a sense, and say that Grandi's series is equal to the "infinitith" term of this new seqeucne 

We have "1,1/2, 2/3,2/4, 3/5, 3/6, 4/7, 4/8, 5/9, 5/10, 6/11,6/12,7/13.....". Because I can't type limit notation and sequence notation on this site, consider what number the numbers in this sequence are getting closer and closer to as you progress along the sequence. When n is even we have 1/2.  Look at the odd terms of the above sequence "1/1 , 2/3, 3/5, 4/7..." the odd terms are getting closer and closer to 1/2. Infinitith term is therefore 1/2. It is from this that you can see:

From the presupposition that ("1+1-1+1-1+1-1+1-1..." to infinity)= ( infinitith term of "1,1/2, 2/3,2/4, 3/5, 3/6, 4/7, 4/8, 5/9, 5/10, 6/11,6/12,7/13....."), 

Grandi's series= 1/2

Infinity isn't a number, so if I was to be formally correct in my argument you would have to use limit notation. But I hope you know what I mean. 

[Hamilcar]

5 hours ago, lmfao said:

@Hamilcar Grandi's series is divergent. As you increase the number of terms of the series you are not getting closer and closer to a particular number, so the fact that the sum=0.5 shouldn't be a brain fuck because 0.5 is arrived at from choosing an arbitrary method of summation. 1+1-1+1-1+1... To  infinity has no real solution but something like 1/2 + 1/4 +1/8 +1/16 + 1/32 +.... To infinity has a far more real solution. Because infinity isn't an actual number (and hence "limits" as a concept are defined) , "addition"is defined as acting different for a convergent series than it is for a divergent series. The way addition is defined for a convergent series is what everyone is used to, you could say that people just decided to invent a different type of addition for summing a divergent series. Here's how 1/2 was the answer for that series you mentioned. 

Let's consider a new sequence for  which the nth term of the new sequence tells you the mean of the first "n" terms of the Grandi's series. E.g., (1-1+1)/3 = 2/3 and (1-1+1-1)=2/4. We are going to redefine addition in a sense, and say that Grandi's series is equal to the "infinitith" term of this new seqeucne 

We have "1,1/2, 2/3,2/4, 3/5, 3/6, 4/7, 4/8, 5/9, 5/10, 6/11,6/12,7/13.....". Because I can't type limit notation and sequence notation on this site, consider what number the numbers in this sequence are getting closer and closer to as you progress along the sequence. When n is even we have 1/2.  Look at the odd terms of the above sequence "1/1 , 2/3, 3/5, 4/7..." the odd terms are getting closer and closer to 1/2. Infinitith term is therefore 1/2. It is from this that you can see:

From the presupposition that ("1+1-1+1-1+1-1+1-1..." to infinity)= ( infinitith term of "1,1/2, 2/3,2/4, 3/5, 3/6, 4/7, 4/8, 5/9, 5/10, 6/11,6/12,7/13....."), 

Grandi's series= 1/2

Infinity isn't a number, so if I was to be formally correct in my argument you would have to use limit notation. But I hope you know what I mean. 

nice explanation...
yes you could average the terms from the first position to the n-th position , and get a convergence towards 1/2 .
there's ramanujan's formula too : S = 1+2+3+4+5... = -1/12
 

[lmfao]

@Hamilcar yeah idk how that other formula works but I think it's safe to say that there some dodgy manipulation which is still referred to as "addition" lol.