Faire des mathématiques c’est donner le même nom a des choses différentes - H. Poincarré / micmath

see also

• Riemann Hypothesis $1^x + 2^x + 3^x + …= 0 ?$
• Numberphile v. Math: the truth about 1+2+3+…=-1/12
  • assert that infinite sum is only defined in correspondance with convergence
  • then as intermediate sum $A$ and $B$ do not converge, then sum does not exists
  • which is kind of redifining what is sum and when it exists.
• $1+2+3+4+…=-1/12$ - is considered as an analytic continuation value, a method in complex analysis that lets you extend the domain of a function beyond where its original definition converges, while preserving its “shape” (analyticity). In this case:
  • the original series diverges at those points
  • but the extended (analytically continued) function assigns finite values there

Demonstration

$1+2+3+4+…=-1/12$

soit $A=1-1+1-1+1-1+…$

$-A = -1+1-1+1-1+…$
$1-A = 1-1+1-1+1-1+…$
donc $1-A=A$

qui donne $A=1/2$

soit $B=1-2+3-4+5-6+7…$
$A+B = 2-3+4-5+6-7+…$
$-1+A+B=-1+2-3+4-5+6-7+…$
donc $-1+A+B=-B$
et $-1+1/2=-2B$

qui donne $B=1/4$

Enfin

soit $C=1+2+3+4+5+6+7+…$
$C-B=4+8+12+16+…$

qui correspond a la somme de la table des 4:
$C-B=4.(1+2+3+4+…) = 4.C$

En remplacant $B=1/4$ on trouve $-1/4 = 3C$
et donc finalement $C=-1/12$

Voila, voila, voila… on est bien avancé…