Sometimes we come across things in our daily life that seems to be incorrect or challenges whatever we have learned and thus we quickly reach to a conclusion that it must be wrong. After reaching this conclusion we are left with only two options.

First option is more obvious and chosen by most of us is, is to continue with our knowledge and the conclusion that we have reached.

Second option is less obvious and involves putting some efforts to at least to try to find out the correctness of our conclusion.

Let’s take an example of one statement, sum of all natural numbers that is 1+2+3+4…, all the way to infinity is equals to -1/12.

After reading this statement, for most of us the conclusion is simple and we won’t even try to consider any other possibility. And that conclusion is it is not possible. Sum of positive number cannot be negative. But few of us will try to see beyond this conclusion and starts questioning. First question is how we even come across this statement which is easily seen to be wrong and that means there must be some reason to have such an equation.

Only take away from this is that whenever we come across such situations, try to go for second option. It might turns out that your conclusion was right and in the process you learn something new. Or better what if your conclusion turns out to be wrong.

Let’s continue with our example.

This sum is known as Ramanujan Summation and is very important in mathematical applications. It states that if you add all the natural numbers, that is 1,2,3,4, and son on, all the way to infinity, you will find that it is equals to -1/12.

What is follow is the proof. But its proof makes use of 2 more series that seem equally interesting and false.

First,

1-1+1-1+1-1+… = ½

Proof,

Let’s assume,

A= 1+1-1+1-1+…

Now subtract A from 1, we get

1-A = 1- (1-1+1+1…), opening brackets

1-A = A => A= ½

Second,

1-2+3-4+5-6…= ¼

Proof,

Let’s assume,

B= 1-2+3-4+5…

Now, subtract B from A, we get

A-B = (1-1+1-1+…)- ( 1-2+3-4…)

= (1-1) + ( -1+2) + (1-3)+..

= 1-2 +3-4…

A-B = B => B=A/2=> ¼

Now, the Ramanujan Summation,

C= 1+2+3+4…

Now, subtract C from B, we get

B-C = (1-2+3-4+5…) – (1+2+3+4+5…)

= (1-1) + ( -2-2) + ( 3-3) + ( -4-4) +…

= 0 -4 +0 -8+…

=-4( 1+2+3+….)

=-4C => C= -B/3=> C= -1/12. Voila

April 21, 2021 at 4:39 pm

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June 4, 2021 at 6:04 pm

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