What do you think the sum of every natural number would be? Most of you would guess infinity. But there exists a summation which plays with numbers which states that all the natural numbers added together instead equals -(1/12)!!!! (Exclamation mark does not mean factorial). . This summation was discovered by Indian mathematician Srinivasa Ramanujan. To understand the Ramanujan Summation, we must understand two other summations which are:
1-1+1-1+1-1+1-1+1-1……. = ½
1-2+3-4+5-6…….. = ¼
The First Statement
Now I’ll show you how the first statement equals ½.
We’ll give the expression a value of A.
So A = 1-1+1-1+1-1+1-1…
Now we do a fun little trick, we subtract A from 1, and we have to subtract the other side of the equation from 1:
1-A = 1-(1-1+1-1+1-1+1-1+1-1…)
Since the part which is on right side of equation is infinite it would look exactly the same. Which is equal to A.
1-A = 1-1+1-1+1-1+1-1+1-1…
1-A = A
Now we do some basic algebra.
1-A+A = A+A We know a negative number plus it’s positive self is always 0. For example -3+3 = 0, like that on the left side of the equation there is -A+A which is 0. And 1 plus 0 is 1, and on the right side of the equation A+A is 2A.
1 = 2A Now we need to get A by itself on the right side so we need to divide 2A by 2 but we also have to do that on the right side. 1 / 2 equals ½ and 2A/A = ½ = A Voila! This is our final statement!! ½ = A! And if you realize, A is equal to that long sequence up there! Hope you enjoyed this part! Now let us move on to the second statement.
The Second Statement
The second statement says that 1-2+3-4+5-6…. = ¼ We’ll call this infinite series B.
B = 1-2+3-4+5-6… Now instead of subtracting B from 1, we’re going to subtract it from A, which is our previous value.
A-B = (1-1+1-1+1-1…) – (1-2+3-4+5-6…) Now we pair up the numbers like this and we see a mind-blowing result:
(1-1) + (-1+2) + (1-3) + (-1+4) + (1-5) + (-1+6)…
= 0+1-2+3-4+5-6…
We get our original series!!!! So we know that A-B = B
So we do some simple algebra:
A-B = B
A = 2B
½ = 2B
¼ = B We got our original value which is B!!!! And found out ¼ is the answer to our second series!!!!
The Ramanujan Summation
Now’s the part all of you are waiting for!!! The Ramanujan Summation!!
The third statement = 1+2+3+4+5+6… = -(1/12)
We’ll give this series the value of C and you probably guessed it, we’re going to subtract C from B.
B-C = (1-2+3-4+5-6…) – (1+2+3+4+5+6…)
Now we can change the 1+2+3+4+5+6… to 1-2-3-4-5-6… by distributing the minus sign. Now we get:
B-C = (1-2+3-4+5-6…) – 1-2-3-4-5-6… Now we can pair up the numbers like we did in the last sum.
(1-1) + (-2-2) + (3-3) + (-4-4) + (5-5) + (-6-6)…
Which equals 0-4 + 0-8 + 0-12… which is equal to -4-8-12… You definitely didn’t expect this! Now, sit tight because this trick will make it all worth it.
I’m going to factor the expression by pulling out -4 since all of the numbers in it are multiples of -4. So the factored expression would look like this:
-4(1+2+3…) We end up with -4 times our original sequence which is C!!!!
B-C = -4C Now we will add C from both sides to make it simpler.
B = -3C We know that B = ¼ so we will substitute that value for B.
¼ = -3C Now we will divide each side by -3 to get C by itself on the right side. And ¼ divided by -3 is 1/-12
-(1/12) = C We got -( 1/12) as our answer which equals C which is our sequence!!!!! Very wonderful and interesting!!! Thank you for reading my blog, I hope you learned something new! As of all, have a nice day!
(Exclamation marks in this case do not mean factorial, they express astonishment or fascination.)
Credits: Cantors Paradise
-written by Vishrut Kinikar
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