The Power of Zero

(Update: A few members have asked me to make this less 'random', more to the point and more readable)

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The Power of Zero OR 

To get somewhere, remember where you came from OR 

Bad Math and Bad Rebuttals

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I recently received a few links ([1], [2], [3], [4]) which show some 'brain-melting maths' to prove that the infinite series represented by S = 1 + 2 + 3 + 4 + 5 + ... is equal to a small negative fraction (- ¹⁄₁₂ to be specific). I am sure that the brain melting preceded the maths rather than the other way around, else this would not have seen the light of day, let alone so many posts.

On the face of it, the proof looks to be quite smart, which explains why it is showing up in so many places. Plus, the honorific attached to the authors was that of Professors, so very few people actually challenged it.

The rebuttal [5] missed the point as well.

First a summary:

The 'proof' uses the Cesaro summation of the Grandi's series as the foundation (*NOTHING* wrong with this, in the context).
Reusing the text from the 'Bad Rebuttal' which actually provides a good summary:
============= Begin Ctrl-c-Ctrl-v ====================
We'll consider three infinite series:

S₁ = 1 - 1 + 1 - 1 + 1 - 1 + ...
S₂ = 1 - 2 + 3 - 4 + 5 - 6 + ...
S₃ = 1 + 2 + 3 + 4 + 5 + 6 + ...

S₁ is something called Grandi's series. According to the video, taken to infinity, Grandi's series alternates between 0 and 1. So to get a value for the full series, you can just take the average - so we'll say that S₁ =  ¹⁄₂. <snip>

Now, consider S₂. We're going to add S₂ to itself. When we write it, we'll do a bit of offset:

S₂          : 1 - 2 + 3 - 4 + 5 - 6 + ...
+ S₂      :      1 - 2 + 3 - 4 + 5 + ...
====   : ===============================
2 * S₂     : 1 - 1 + 1 - 1 + 1 - 1 + ...

So 2 * S₂ = S₁; therefore S₂ = ^S₁⁄₂= ¹⁄₄.

Now, let's look at what happens if we take the S₃, and subtract S₂ from it:

S₃           :    1 + 2 + 3 + 4 + 5 + 6 + ...
- S₂         : - [1 - 2  + 3 - 4  + 5  - 6 + ...]
===== : ===============================
S₃ - S₂    :    0 + 4 + 0 + 8 + 0 + 12 + ...
or
S₃ - S₂     : = 4(1 + 2 + 3 + ...) = 4 * S₃ and therefore -S₂ = 3 * S₃ and S₃ = -¹⁄₁₂.

======== End Ctrl-c-Ctrl-v and Bad-Maths ==========

Good Maths - The actual rebuttal, etc:

First, there is nothing wrong with assigning the Cesaro summation of ¹⁄₂ to Grandi's series, as long as you know 'where you came from'.

Understand it this way, if a car starts at rest, and then accelerates at a constant rate to hit 1 mile per hour (a rather safe driver) in exactly one hour, and then decelerates at the same constant rate in the next hour, his speed at the hour mark would be 1, then 0, then 1 again and so on. This is what you expect at the end of each term in the Grandi series (n₁:1, n₂:1-1=0, n₃:1-1+1=1, and so on). By following the above approach, the car would have covered ¹⁄₂ mile in the first hour, and another ¹⁄₂ mile in the second. If the driver continues in the same fastidious vein till gas stocks last, he would have done an 'average' of ¹⁄₂ mile per hour over the course of the adventure. You can also think of this as the area under the line graph that keeps going from 0 to 1 and then back to 0 after each unit on the x-axis. Phil Plait uses a similar Stairways to Heaven analogy in link [4] - look at that graph for reference or the first graph on the Google Spreadsheet [6] [http://goo.gl/jtbZi2]. 

 S₂ derivation is also correct as the average converges to ¹⁄₄:

Now that we have the Cesaro sums out of the way, we are ready to identify the fallacy - remember the power of the zero!

In fact, the original proof is correct up to this point:
S₃ - S₂: 0 + 4 + 0 + 8 + 0 + 12 + ...

The failure is in the next step where the right hand side is equated to 4 times S₃. The power of zero resides in the fact that you cannot ignore it. Remember, the car analogy up top? If the car runs at a constant speed for an hour, it *seriously* impacts the averages, and it does not actually start running 4 times faster if it does a 4x speed only the other hour. Fact is, the right hand side of the equation is actually marginally less than S₃. It would be closer to state that S₃ - S₂ = ~S₃ (approximately S₃) or almost S₃ if our driver broke all laws of physics (and the speed of light barrier) and followed the speed-acceleration requirement of the higher terms in our series. Look at the graph 3 in [6], Average of (S₃ - S₂₎ (red line) is marginally lower than Average of S₃ (orange line).

Graph 4 compares S₃ (Green line) and (S₃ - S₂₎ (Blue line), you can see how S₃ manages a slender lead while the Blue line zigs-zags around it.:

The 'trick' or mistake lies in forgetting 'where we came from' - Cesaro sum makes sense for the series, if and only if you are looking at the 'averaging approximations'. Once you start doing mathematical operations with infinity, the vortex opens and sucks your melted brain into the parallel universe.

However, if forced to approach the Cesaro sums with arithmetics, do note that by the very nature of partial sums, the order of the series and the presence of zeroes may make a huge difference to the partial sums. Also, deductions on series where the 'average' is a divergent function, require special care and caution. Focusing on the differential of the the function should make arithmetic more intuitive and also allow one to appreciate the impact of zero slope.

So long, and thank you for the fish, er your time, for reading this. Hope you learnt something about the dependability of whole numbers - their sum does NOT become negative, however you may twist it - AND any 'Physics application' of this 'property', as claimed by the Wikipedia article [7], needs a serious peer review.

I think Ramanujan was pranking Dr Hardy in 1913 with this.

Cheers,
Rahul Raj
Not-a-mathematician-in-spite-of-my-teachers'-best-efforts

Links:

Bad Maths:
[1] Authors' site - 'Original' source : http://www.numberphile.com/videos/analytical_continuation1.html
[2] On YouTube - Same source: http://www.youtube.com/watch?v=w-I6XTVZXww#t=22
[3] Authors' blog where they defend the same : http://periodicvideos.blogspot.co.uk/2014/01/thanks.html
[4] Bad Astronomer - Phil Plait not understanding the less celestial numbers/ the FB link I received through multiple sources: http://www.slate.com/blogs/bad_astronomy/2014/01/17/infinite_series_when_the_sum_of_all_positive_integers_is_a_small_negative.html

Bad Rebuttal:
[5] Someone who 'misses' the point : http://scientopia.org/blogs/goodmath/2014/01/17/bad-math-from-the-bad-astronomer/

Almost Good Maths:
[6] Calculations for the Cesaro sum - Google Docs link: http://goo.gl/jtbZi2

Wiki:
[7] http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_⋯