Is the sum of all natural numbers = -1/12? Part Two

stormblaze (62)in #mathematics • 8 years ago

Ok here is part two of "Is the sum of natural numbers = -1/12?" It is time to uncover this result and find the beauty of maths!

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In the first part I showed you how to calculate a convergent sum and then I demonstrated how to calculate a divergent sum! If you missed the first part click here . Now it’s time to get our hands l dirty and calculate the sum of all natural numbers. You need to keep a mathematical open mind and please try to keep in mind that everything being done is arithmetically correct!

Let us begin :

If you use the same technique as shown in the previous post, You will see that we can calculate this strange series which we will call Sx:

Sx =1-1+1-1+1-1…

Now this looks crazy but the answer of this sum is = 1/2 …

Sx = ½

You can use the technique in the previous post and calculate the answer for fun, there are many ways to find it as well!

Now let us continue with the hard work and find the answer of our age old question, Is the sum of natural numbers = -1/12?

In order to find out we must define another sum which we will call Sb:

Sb = 1-2+3-4+5….

Now we make add Sb With itself and add zero like this:

Sb =1-2+3-4+5…
+Sb+0=0+1-2+3-4+...

So now we get :

2Sb= 1-1+1-1+...

We see 2Sb = Sx So:

2Sb = ½

If we simplify we get:

Sb = ¼

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Ok we are almost at the finish line, here comes the last stretch:

If we define Sn as the sum of all natural numbers and subtract Sb we get the following:

Sn = 1+2+3+4+5
-Sb = -[1-2+3-4+5-...]

We get :

Sn - Sb = 0 +4+0+8+0+12+...
Sn - Sb = 4+8+12+....

So If we take out a factor 4 we get :

Sn-Sb = 4[1+2+3+4+....]

So we can write :

Sn-Sb = 4Sn

We also know what the value of Sb so:

Sn-¼ = 4Sn

Now we simplify :

-3Sn = ¼

And simplify even more:

Sn = -1/12!

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The Verdict:

Ok there you have it! It is just so ridiculous! But I love this result. We do not know why the sum of all natural numbers is a negative fraction, but we see it all over natural phenomena like quantum mechanics. But it is important to note that in math we have things called axioms. An axiom is a statement in a beginning of a math proof or paper which states what you consider as to be a mathematical truth and you use it to build your proof on! For example:

You can say the square root of -1 is i.

If you ever get a square root of - 1 You would know that you are allowed to say i. If you did not state it in your axioms, then it means there is no answer.

So basically what I am saying is, the result is in the eye of the axiom. You can state in your axioms that there is no way to calculate the sum of a divergent series, well then our result -1/12 does not apply. You can also do the opposite and find that the sum of all natural numbers is - 1/12 .

That is the beauty of math. You can define your own world where this laws apply and find the answers of that world!

Thank you for reading this long post hoped you enjoyed! Give an upvote, leave a comment below of what you think!

And as always

Thanks for reading!

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