极限系列考研题

设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

原始限额=s(1)

=a/(a-1)^2