Ramanujan summation Series

Question

I am a class 12 student and I am Wondering about the Ramanujan summation Series that is 1+2+3+..... = -1/12. As per my knowledge Summation of positive number gives a positive number but in the Ramanujan summation Series value is negative. Can you please explain this in brief.

Answer 1

Some properties of addition can be proven for any finite number $n$ of terms by induction on $n$. If I mention that the sum of $n$ integers is an integer, or (to take an example mathematicians care more about) the sum of $n$ integers doesn't depend on how you order them, (a) you know I'm right and (b) if you tried writing down a proof, it'd use induction. But just because any finite size for the list of summed terms achieves such obvious outcomes, it doesn't mean infinite lists do, because induction can't be extended to those.

In fact, it turns out there isn't one unique way to define the sum of infinite series that reproduces the usual sum of finitely many terms in the case where only finitely many terms in the infinite series are non-zero. Ramanujan summation isn't the most widely used definition. The most widely used one is the $n\to\infty$ limit of the sum of the first $n$ terms, which in this case would give $\infty$ instead of $-1/12$. (Note $\infty$ isn't an integer either!)

But even the usual limit-of-partial-sums definition can have surprising outcomes, in the sense of induction-provable results failing. For example, the sum of finitely many rational numbers is rational; but every irrational number is expressible as a sum of infinitely many rational numbers (e.g. add multiples of powers of $10$ to gradually give the desired limit's decimal digits), so a sum of infinitely many rational numbers needn't be rational.