Manipulating Infinite Sums

!THE FOLLOWING MATH IS HIGHLY ILLOGICAL! Hey guys, I've been talking to my friends Josh Speckman and Nate Ji

So, you've probably seen the 1+2+3+4+5=1121+2+3+4+5 \cdots = - \dfrac{1}{12}. If not, here: We will define three sums: S1=11+11+11S _1 = 1-1+1-1+1-1 \cdots, S2=12+34+56S _2 = 1-2+3-4+5-6 \cdots, and S3=1+2+3+4+5S _3 = 1+2+3+4+5 \cdots.

Looking at the first sum, we see that 1S1=1(11+11+1)=11+11+111-S _1=1-(1-1+1-1+1 \cdots) = 1-1+1-1+1-1 \cdots. Thus, S1=12S _1= \dfrac{1}{2}.

Now we must solve the second sum. We get S2+S2=2S2=12+34+5+12+34+5=1+(2+1)+(32)+(4+3)=11+11+11=12S _2 +S _2 = 2S _2 = 1 -2+3-4+5 \cdots + 1 - 2 + 3 - 4 + 5 \cdots = 1 + (- 2 + 1) + (3-2) + (-4+3) \cdots = 1-1+1-1+1-1 \cdots = \dfrac{1}{2}. Thus 2S2=12S2=142S _2 = \dfrac{1}{2} \Rightarrow S _2 = \dfrac{1}{4}.

Now the last sum. We see that S3S2=1+2+3+4+5(12+34+56)=(11)+(2+2)+(33)+(4+4)=4+8+12+16=4S3S _3 - S _2 = 1+2+3+4+5 \cdots - (1 - 2 + 3 - 4 + 5 - 6 \cdots) = (1-1) + (2+2) +(3-3) + (4+4) \cdots = 4 + 8 + 12 + 16 \cdots = 4S _3. So S314=4S33S3=14S3=112S_3 - \dfrac{1}{4} = 4S _3 \Rightarrow 3S _3 = - \dfrac{1}{4} \Rightarrow S _3 = - \dfrac{1}{12}.

Now, I was showing my friends this, and they decided to continue along the theme of not-quite valid mathematics. So, here it is:

1+2+3+4+51+2+3+4+5\cdots is the sum of the first \infty positive integers, so it is (+1)2=2+2\dfrac{\infty(\infty + 1)}{2} = \dfrac{\infty^2 + \infty}{2}. Since this is equal to 112-\dfrac{1}{12}, we get 2+2=112\dfrac{\infty^2+\infty}{2} = - \dfrac{1}{12},

so 2+=162++16=0\infty^2+\infty=-\dfrac{1}{6} \Rightarrow \infty^2+\infty+\dfrac{1}{6} = 0 ,

and =1±132=3±36\infty = \dfrac{-1 \pm \sqrt{\dfrac{1}{3}}}{2} \Rightarrow \infty =\boxed{ \dfrac{-3 \pm \sqrt{3}}{6}}.

Yay infinite divergent series!

Note by Frodo Baggins
5 years, 3 months ago

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I DON'T UNDERSTAND YOU.

Tanisha Martheswaran - 5 years, 3 months ago

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Well, infinity squared is still infinity and there are different types of infinity so you are practically stating that the divergent sum (zeta of -1) is equal to infinity AND minus one twelfth, and both things can be true.

Bogdan Simeonov - 5 years, 3 months ago

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