Monday, September 4, 2017

sequences and series - The sum of all the odd numbers to infinity

We have this sequence:



S1: 1+2+3+4+5+6.. (to infinity)




It has been demonstrated, that S1 = -1/12.



Now, what happens if i multiply by a factor of 2?



S2: 2+4+6+8+10+12.... (to infinity).



I have 2S1, which is equal to -1/6



On this, we can create a equation for the odd numbers:




S3: 1+3+5+5+7+9+11... (to infinity)



We know that for every term in S2, every term in S3 is just (n-1)



Or, The sum of the even numbers, Minus , the sum of infinitely many (-1)s



So S3 = -1/6 - ∞



However, we also know that the odd numbers + the even numbers = The natural numbers.




So let's try it.



-1/6 - (-1/6 -∞)



We have -1/6 + 1/6 + ∞



Which is just ∞



So, there we have it. a paradox. S1 cannot be both -1/12 or ∞

No comments:

Post a Comment

analysis - Injection, making bijection

I have injection $f \colon A \rightarrow B$ and I want to get bijection. Can I just resting codomain to $f(A)$? I know that every function i...