I have seen a ridiculous "proof" claiming that $$\displaystyle\sum_{n=1}^{\infty} n=-1/12.$$ The starting point is a false statement that
$$\sum_{n=1}^{\infty} (-1)^{n+1}=1/2.$$
Since we know that $\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1}$ diverges, any argument based on the value of it will be invalid.
But if we look at this from a different angle, am wondering if the following makes sense:
For any series $\displaystyle\sum_{n=1}^{\infty} a_n$, let its "mass" (for lack of a better terminology at this moment) be a real number $m\left(\displaystyle\sum_{n=1}^{\infty} a_n\right)\in\mathbb{R}$ such that the following hold:
- If $\displaystyle\sum_{n=1}^{\infty} a_n$ converges, then $m\left(\displaystyle\sum_{n=1}^{\infty} a_n\right)=\displaystyle\sum_{n=1}^{\infty} a_n$.
- If the sequence $\{b_n\}$ is obtained from the sequence $\{a_n\}$ by inserting or deleting a list (possibly infinite) of $0$'s, then $m\left(\displaystyle\sum_{n=1}^{\infty}a_n\right)=m\left(\displaystyle\sum_{n=1}^{\infty} b_n\right)$.
- $m\left(\displaystyle\sum_{n=1}^{\infty} (a_n+b_n)\right)=m\left(\displaystyle\sum_{n=1}^{\infty} a_n\right)+m\left(\displaystyle\sum_{n=1}^{\infty} b_n\right)$.
- If $C$ is a real number, $m\left(\displaystyle\sum_{n=1}^{\infty} (Ca_n)\right)=C\times m\left(\displaystyle\sum_{n=1}^{\infty} a_n\right)$.
- If the sequence $\{b_n\}$ is obtained from the sequence $\{a_n\}$ by reordering finitely many terms, then $m\left(\displaystyle\sum_{n=1}^{\infty}a_n\right)=m\left(\displaystyle\sum_{n=1}^{\infty} b_n\right)$.
- $m\left(\displaystyle\sum_{n=1}^{\infty} (-1)^{n+1}\right)=1/2.$
Then will there be any contradictions among the above axioms? Can we uniquely determine the mass of any series? For example, can one prove that
$$m\left(\sum_{n=1}^{\infty} n\right)=-1/12?$$
Furthermore, if we replace the sixth condition by the following:
$$m\left(\sum_{n=1}^{\infty} (-1)^{n+1}\right)=k,$$
where $k$ is a fixed real number, can we uniquely determine the mass of any series?