Supongamos s (x) = ∑ {1,∞} n * x (n-1)/a n.
=∑{0,∞}(n+1)*x^n/a^(n+1)
∫{0,x}s(x)dx= ∫{0,x}[∑{0,∞}(n+1)*x^n/a^(n+1)]dx
=∑{0,∞}[∫{0 ,x}(n+1)*x^n/a^(n+1)dx]
=∑{0,∞}(x/a)^(n+1)
=x/a*∑{0,∞}(x/a)^n
=x/a*1/(1-x/a)
=x/(a-x)
s(x)=[x/(a-x)]'=a/(a-x)^2
Límite original=s(1)
=a/(a-1)^2