Hyperbola Fermat, Geometric Infinite Sum

[petroljose]

Hello everybody,

I'm trying to understand some steps in the evolution of calculus, and in a .pdf found in the internet I read the document: http://www.ugr.es/~mmartins/old_web/Docencia/Old/Docencia-Matematicas/Historia_de_la_matematica/clase_3-web.pdf , in pags. 14-15. I want to solve the to equations of the two areas (1/ar) and (r/a), but I when I try solving the infinite sum inside the symbols: Ʃ, for example, with the first term for 1/r^k for k=0 equal to 1, I don't reach the same resoult (1/ar). So anybody can help me to understand this, and why lastly the area isn't 1/ar nor r/a, but 1/a as Fermat proved.

Thank you all.

 

[petroljose]

Misprint in the link

Excuse me, the link is: http://www.ugr.es/~dpto_am/OLD/docencia/Apuntes/Origenes_del_Calculo.pdf

 

[petroljose]

Posible solution

With the formula to sum the infinite terms of a geometric series, I solve as in the figure attached.

 

[petroljose]

Solved:

Only simplifying:( (r-1)/(a*r^2 ) )*(1-1/r) by multiplying the sum 1/(1-1/r) by r so: r/(r-r/r)=r/(r-1) and then:
((r-1)/(a*r^2))*(r/(r-1))=1/ar