number theory - How does one get that $1^3+2^3+3^3+4^3+cdots=frac{1}{120}$?

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.

Answer

I think I have rediscovered it (after watching the video you linked and read wiki biography of Ramanujan).

Start with

$$\frac{1}{x+1} =1 -x +x^2 -x^3 +-\cdots, \quad |x| <1.$$ and differentiate to get $$-\frac{1}{(x+1)^2} =-1 +2x -3x^2 +4x^3 -+\cdots, \quad |x| <1. \\ \frac{2}{(x+1)^3} =2 \cdot 1 -3 \cdot 2x +4 \cdot 3 x^2 -+\cdots, \quad |x| <1. \\ -\frac{6}{(x+1)^4} =-3 \cdot 2 \cdot 1 +4 \cdot 3 \cdot 2x -5 \cdot 4 \cdot 3 x^2 +-\cdots, \quad |x| <1.$$

Take a magic mushroom and, ignoring $|x| <1$, let us take $x=1$ in each one.

$$\frac{1}{2} =1 -1 +1 -1 +- \cdots \\ -\frac{1}{4} =-1 +2 -3 +4 -+ \cdots \\ \frac{1}{4} =2 \cdot 1 -3 \cdot 2 +4 \cdot 3 -+ \cdots \\ -\frac{3}{8} =-3 \cdot 2 \cdot 1 +4 \cdot 3 \cdot 2 -5 \cdot 4 \cdot 3 +- \cdots$$ Or more formally, $$\sum_{m=1}^\infty (-1)^{m+1} m =\frac{1}{4}. \\ \sum_{m=1}^\infty (-1)^{m+1} m (m+1) =\frac{1}{4}. \\ \sum_{m=1}^\infty (-1)^{m+1} m (m+1) (m+2) =\frac{3}{8}.$$

But notice

$$\begin{align} \sum_{m=1}^\infty (-1)^{m+1} m^2 =&\sum_{m=1}^\infty (-1)^{m+1} m (m+1) -\sum_{m=1}^\infty (-1)^{m+1} m \\ =&\frac{1}{4} -\frac{1}{4} =0. \end{align}$$ and $$\begin{align} \sum_{m=1}^\infty (-1)^{m+1} m^3 =&\sum_{m=1}^\infty (-1)^{m+1} m (m+1) (m+2) -3\sum_{m=1}^\infty (-1)^{m+1} m^2 -2\sum_{m=1}^\infty (-1)^{m+1} m \\ =&\frac{3}{8} -3 \cdot 0 -2 \cdot \frac{1}{4} =-\frac{1}{8} \quad \quad \ldots \spadesuit \end{align}$$

On the other hand,

$$\zeta(-3) :=1^3 +2^3 +3^3 +\cdots \\ 2^4 \zeta(-3) =2 \cdot 2^3 +2 \cdot 4^3 +2 \cdot 6^3 +\cdots \\ $$ Subtract them, aligning the 2nd, 4th, 6th term like Ramanjunan did in his notebooks (shown in the video). $$-15 \zeta(-3) =1^3 -2^3 +3^3 -+\cdots \quad \quad \ldots \heartsuit$$ $\heartsuit$ and $\spadesuit$ together give us: (Hold your breath.) $$\sum_{m=1}^\infty m^3 =\frac{1}{120}.$$

Recently I also found a proof of Riemann conjecture, but the answer box is too narrow for me to type all that down.

P.s. seriously, I think Ramanujan's effort is sort of finding an interpretation of divergent series so that they have a real value, while their manipulation to be still consistent to our usual notion of arithmetics: arranging, addition, expanding, etc.?
Maybe this can be compared to the attempt to define quaternion as an extension of complex numbers, while inevitably discarding commutative law?