# I 1+4+9+16+...=0 proof

1. Jun 14, 2016

### Qemikal

Hello, I started to learn divergent series/sums, to practice I calculated some basic ones, you know: 1+2+3+4+5+6...= -1/12, but I really had problems when i tried to demonstrate that 1+4+9+16+...= 0(the sum of squares of natural numbers), I've tried to add, subtract etc, but I couldn't prove it, anyone here could help?

2. Jun 14, 2016

### Staff: Mentor

I guess you are referring to things like this? Forget it. Those calculations are mathematically nonsense, and you can get any result you like if you use those wrong operations. They are not the way those pseudo-limits are defined.

As an example, consider
1+1+1+1+.... = X
Let's add 1 to both sides:
1+1+1+1+... = 1+X
But the left sides of both equations are identical, therefore, X=1+X.
Subtract X:
0=1.

3. Jun 14, 2016

### Qemikal

S= 1-2+3-4+5... And adding S+S, it'll eventually give us 2*S=1-1+1-1+...
That's how we can assign values to these series => S= 1/4 (1-1+1-1+1...= 1/2 )

4. Jun 14, 2016

### PeroK

What about $1 + 4 + 9 + 16 \dots = 1 +(-1 + 1) + 4 + (-5 + 5) + 9 + (-14 + 14) + 16 + (-30 + 30) \dots = 0 + 5 - 5 + 14 - 14 + 30 - 30 + \dots = 0 + 0 + 0 + 0 \dots = 0$

Although:

$1 + 2 + 3 + 4 + 5 \dots = 1 + (-1 + 1) + 2 + (-3 + 3) + 4 + (-7 + 7) + 5 + (-12 + 12) \dots) = 0$

Looks like they all sum to 0.

5. Jun 14, 2016

### Staff: Mentor

You can, but those values are meaningless. See the example I put in my post, you can easily make contradictions like that.

It is not what mathematicians do to assign values to those series!

6. Jun 14, 2016

### Qemikal

I wrote that I add sums, in general, mathematicians do that, but in this example you added 1 to an infinite series that's wrong, infinity + 1 = infinity.

As written in that wikipedia article mathematician 'assign' the values to series, maybe equal is much said, but how to assign 0 to the 1+4+9+... would be a better formulation of the question.

This method of assigning is used in many papers(including string theory).
A video from Numberphile would explain visually what I mean:
but I would like to know how to do this for 1+4+9+... series.

7. Jun 14, 2016

### ShayanJ

See this series of videos!
I don't remember in which, but he does explain assigning values to divergent series.

8. Jun 14, 2016

### Staff: Mentor

Not in the way you wrote here, no.
That is my point, the operations you do with the series are wrong.
There are various methods, in this particular case it is Ramanujan summation. More options (which can lead to different answers for the same series) are listed here.

9. Jun 14, 2016

### Qemikal

How would be a correct way to write it?

I would like to know the steps he applied to be able to assign 0 to the middle series.
[PLAIN]http://QUOTE="mfb, [Broken] post: 5498284, member: 405866"]Not in the way you wrote here, no[/QUOTE] [/PLAIN]

Last edited by a moderator: May 8, 2017
10. Jun 14, 2016

### Staff: Mentor

See the section above, "Summation".

11. Jun 14, 2016

### Staff: Mentor

As mfb said,
This misuse of divergent series has been discussed numerous times here at PF.