Man who knew infinity

Recipy that disturbed the fundamentals of Mathematics

Here is blogs first post , this is about the fact that shook my fundamentals to the core in mathematics while i was preparing for comptetive exams

Sum of n natural Number{N} is given as Nn = (n(n+1)/2) (Summation is Denoted as N )

as n -> infinity for Sn , sum of all natural Numbers is given as

    Nn = 1 + 2 + 3 + 4 ---- inf = -1/12

To prove this statement or to even understand it ,mathematicians had to go through so much pain

Let us discuss how this actually efected the reality of our understanding

Here I am providing simple explanation for understanding , for complex one in future readings … (Reiman Zeta)

Proof

at Initial

Take a Geometric Series

Sn = 1-1+1-1+1-1+... Inf   (Eq1)


Sn = 1-(S1)

2Sn=1

Sn = 1/2

If you see the Eq1(Grandis Series) we expect the answer to be zero or one by logic , but we get Sn=1/2 even though Sn is divergent series

Geometric series 1/1 + 1/2 + 1/4 + 1/8 .. = 2 (Here it converges to 2 ) where as 1 + 2 + 3 + 4 + 5 .. is always increasing , this is known as divergent . 1-1+1-1+1-1 …. is also a divergent series , answer could be {1,0} depends on where you stop but in infinity world it is divergent

Another similar series

Sn = 1-2+3-4+5-6+...        (Eq2)  

2 Sn = (1-2+3-4+5-6+... ) + (0+1-2+3-4+5-6+...)
2 Sn = 1-1+1-1+1-1...
2 Sn = 1/2
Sn = 1/4

Actual Question :

  • Sum of infinity natural numbers i.e., Sn = 1+2+3+4+5+6…
let Y = 1-2+3-4+5-6+... 
Sn - Y = 4 + 8 + 12 + 16 + ...
Sn - Y = 4(1+2+3+4+...)
Sn - Y = 4Sn
Sn - 1/4 = 4Sn (from Eq2)
-3Sn = 1/4
Sn = -1/12

End Thoughts :