The man who knew infinity I mean Srinivasan Ramanujan the most brilliant mind in the history of mathematics is known for the various mathematical proofs which he had given to the world, out of which the most famous one is sum of all natural number till infinity for which Ramanujan had entitled with the word infinity.For those of you who are unfamiliar with this series, which has come to be known as the Ramanujan Summation after a famous Indian mathematician named Srinivasa Ramanujan, it states that if you add all the natural numbers, that is 1, 2, 3, 4, and so on, all the way to infinity, you will find that it is equal to -1/12. Yup, -0.08333333333.
Don’t believe me? Keep reading to find out how I prove this, by proving two equally crazy claims:
1-1+1-1+1-1 ⋯ = 1/2
1-2+3-4+5-6⋯ = 1/4
First off, the bread and 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 to 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 do 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, do a bit of high school algebra and 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, and while it is my personal favourite, there isn’t a cool history or discovery story behind this. However, it does open the door to proving a lot of interesting things, including a very important equation for quantum mechanics and 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 play around with it. This time, instead of subtracting B from 1, we are going to 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, and 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, and even helped extend Euler’s research in the Basel Problem and lead towards important mathematical functions like the Riemann Zeta function.
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⋯, and you may have been able to guess it, we are going to 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⋯
B-C = -4-8-12⋯
Not what you were expecting right? Well hold on to your socks, because I have one last trick up my sleeve that is going to 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, and 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 and we get our magical result:
1/4 = -3C
1/-12 = C or C = -1/12
Now to know more watch this video till the end.
Thanks for watching..
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क्यों इंफिनिटी से डरते थे Ramanujan | Ramanujan vs infinity | The man who knew infinity
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5 окт 2024
