Sum of all natural numbers

Yes, It’s not the most exciting title for a blog post. Lets change it to “is this possible that  sum of natural numbers can be equal to fraction, negative fraction, like -1/12?”.

Once we dealt with title we can move on to the subject, first how you define a sum of all natural numbers?

all natural numbers are respectively:

1+2+3+4….+∞

Now, How to define a sum of those numbers:

In mathematics we write:

\sum_{k=1}^n k = \frac{n(n+1)}{2},

For normal people this  doesn’t mean anything. So let’s take a look at the visual interpretation of this  series.

 

re

 

Here we can see that this series goes to infinity and don’t have any limits.

 

Because the sequence of partial sums fails to converge to a finite limit, the series does not have a sum.

 

Regardless of this some mathematicians, like Ramanujan, managed to extract meaningful values from this numbers.

Lets use method used in Numberphile’s video:
// Value we are looking for
S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + … = ?
// Sum of this is equal to 1/2 this is done by not complex transformations

S1 = 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + … = 1/2

//From this point we can continue to
S2 = 1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + …
2S2 = 1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + …
+ 1 − 2 + 3 − 4 + 5 − 6 + 7 + …
= 1 − 1 + 1 − 1 +  1 −  1 +  1 − 1 + … = 1/2
S2 = 1/4

//And finally we can go to:
S − S2 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + …
− 1 + 2 − 3 + 4 − 5 + 6 − 7 + 8 + …
= 0 + 4 + 0 + 8 + 0 + 12 + 0 + 16 + … = 4S

S-S2=4S

//using s2 value from last example we end up on:
S – 1/4 = 4S ⇒ S = – 1/12

So is this means that sum of this numbers is equal to -1/12? Answer is, no. It’s because this method(zeta function regularization) only assign values to the series. That mean this number is indeed strongly associated but not equal. That doesn’t  make it less important. This value is commonly used in areas of Physics like quantum physic. That’s it for today. Stay Awesome!