Sum of n, n², or n³

     

This is what my mom said to me when I told her about this little mathematical anomaly. & it is just that, an anomaly. After all, it defies basic logic. How could adding positive numbers equal not only a negative, but a negative fraction? What the frac?

Before I begin: It has been pointed out khổng lồ me that when I talk about sum’s in this article, it is not in the traditional sense of the word. This is because all the series I deal with naturally vị not tend lớn a specific number, so we talk about a different type of sums, namely Cesàro Summations. For anyone interested in the mathematics, Cesàro summations assign values lớn some infinite sums that bởi not converge in the usual sense. “The Cesàro sum is defined as the limit, as n tends lớn infinity, of the sequence of arithmetic means of the first n partial sums of the series” — Wikipedia. I also want to say that throughout this article I khuyến mãi with the concept of countable infinity, a different type of infinity that đơn hàng with a infinite set of numbers, but one where if given enough time you could count to any number in the set. It allows me to use some of the regular properties of mathematics lượt thích commutativity in my equations (which is an axiom I use throughout the article).

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Srinivasa Ramanujan (1887–1920) was an Indian mathematician
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Don’t believe me? Keep reading to lớn find out how I prove this, by proving two equally crazy claims:

1–1+1–1+1–1 ⋯ = 1/21–2+3–4+5–6⋯ = 1/4

First off, the bread và butter. This is where the real magic happens, in fact the other two proofs aren’t possible without this.

I start with a series, A, which is equal lớn 1–1+1–1+1–1 repeated an infinite number of times. I’ll write it as such:

A = 1–1+1–1+1–1⋯

Then I vì chưng a neat little trick. I take away A from 1

1-A=1-(1–1+1–1+1–1⋯)

So far so good? Now here is where the wizardry happens. If I simplify the right side of the equation, I get something very peculiar:

1-A=1–1+1–1+1–1+1⋯

Look familiar? In case you missed it, thats A. Yes, there on that right side of the equation, is the series we started off with. So I can substitute A for that right side, bởi a bit of high school algebra và boom!

1-A =A

1-A+A=A+A

1 = 2A

1/2 = A

This little beauty is Grandi’s series, called such after the Italian mathematician, philosopher, and priest Guido Grandi. That’s really everything this series has, và while it is my personal favourite, there isn’t a cool history or discovery story behind this. However, it does mở cửa the door lớn proving a lot of interesting things, including a very important equation for quantum mechanics và even string theory. But more on that later. For now, we move onto proving #2: 1–2+3–4+5–6⋯ = 1/4.

We start the same way as above, letting the series B =1–2+3–4+5–6⋯. Then we can start to lớn play around with it. This time, instead of subtracting B from 1, we are going khổng lồ subtract it from A. Mathematically, we get this:

A-B = (1–1+1–1+1–1⋯) — (1–2+3–4+5–6⋯)

A-B = (1–1+1–1+1–1) — 1+2–3+4–5+6⋯

Then we shuffle the terms around a little bit, và we see another interesting pattern emerge.

A-B = (1–1) + (–1+2) +(1–3) + (–1+4) + (1–5) + (–1+6)

A-B = 0+1–2+3–4+5⋯

Once again, we get the series we started off with, and from before, we know that A = 1/2, so we use some more basic algebra and prove our second mind blowing fact of today.

A-B = B

A = 2B

1/2 = 2B

1/4 = B

And voila! This equation does not have a fancy name, since it has proven by many mathematicians over the years while simultaneously being labeled a paradoxical equation. Nevertheless, it sparked a debate amongst academics at the time, và even helped extend Euler’s research in the Basel Problem & lead towards important mathematical functions lượt thích the Riemann Zeta function.

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Now for the icing on the cake, the one you’ve been waiting for, the big cheese. Once again we start by letting the series C = 1+2+3+4+5+6⋯, và you may have been able lớn guess it, we are going to lớn subtract C from B.

B-C = (1–2+3–4+5–6⋯)-(1+2+3+4+5+6⋯)

Because math is still awesome, we are going to rearrange the order of some of the numbers in here so we get something that looks familiar, but probably wont be what you are suspecting.

B-C = (1-2+3-4+5-6⋯)-1-2-3-4-5-6⋯

B-C = (1-1) + (-2-2) + (3-3) + (-4-4) + (5-5) + (-6-6) ⋯

B-C = 0-4+0-8+0-12⋯

Not what you were expecting right? Well hold on lớn your socks, because I have one last trick up my sleeve that is going khổng lồ make it all worth it. If you notice, all the terms on the right side are multiples of -4, so we can pull out that constant factor, và lo n’ behold, we get what we started with.

B-C = -4(1+2+3)⋯

B-C = -4C

B = -3C

And since we have a value for B=1/4, we simply put that value in và we get our magical result:

1/4 = -3C

1/-12 = C or C = -1/12

Now, why this is important. Well for starters, it is used in string theory. Not the Stephen Hawking version unfortunately, but actually in the original version of string theory (called Bosonic String Theory). Now unfortunately Bosonic string theory has been somewhat outmoded by the current area of interest, called supersymmetric string theory, but the original theory still has its uses in understanding superstrings, which are integral parts of the aforementioned updated string theory.

The Ramanujan Summation also has had a big impact in the area of general physics, specifically in the solution to lớn the phenomenon know as the Casimir Effect. Hendrik Casimir predicted that given two uncharged conductive plates placed in a vacuum, there exists an attractive force between these plates due to lớn the presence of virtual particles bread by quantum fluctuations. In Casimir’s solution, he uses the very sum we just proved to mã sản phẩm the amount of energy between the plates. Và there is the reason why this value is so important.

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So there you have it, the Ramanujan summation, that was discovered in the early 1900’s, which is still making an impact almost 100 years on in many different branches of physics, and can still win a bet against people who are none the wiser.

P.S. If you are still interested & want to read more, here is a conversation with two physicists trying lớn explain this crazy equation và their views on it’s usefulness và validity. It’s nice & short, and very interesting. Https://physicstoday.scitation.org/do/10.1063/PT.5.8029/full/


This essay is part of a series of stories on math-related topics, published in Cantor’s Paradise, a weekly Medium publication. Thank you for reading!