TITLE: How does one get that $1^3+2^3+3^3+4^3+\cdots=\frac{1}{120}$?
QUESTION [7 upvotes]: While watching interesting mathematics videos, I found one of the papers  of Srinivasa Ramanujan to G.H.Hardy in which he had written $1^3+2^3+3^3+4^3+\cdots=\frac{1}{120}$.
The problem is that every term on the left is more than $\frac{1}{120}$ yet the sum is $\frac{1}{120}$. How is that ???
I know that there are many and much more interesting things  presented by Ramanujan (like $1-1+1-1+1...=\frac{1}{2}$ and $1+2+3+4.....=\frac{-1}{12}$) but for now I am interested in the summation in the title.
Any idea/hint is heartily welcome. Thanks.
Here is the video I'm talking about.

REPLY [3 votes]: The problem is that every term on the left is more than $\frac{1}{120}$ yet the sum is $\frac{1}{120}$. How is that ???

To answer the question directly: Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series. It is well explained in the linked Wikipedia article.