Question

如何求1+1/(2x)+1/(3x^2)+....+1/[n*x^(n-1)]的极限？

1+1/(2x)+1/(3x^2)+....+1/[n*x^(n-1)]

Answer 1

Sum[1/(n x^(n - 1)), {n, 1, Infinity}]

Answer 2

解：令S(x)=∑(n=1->∞) 1/nx^(n-1)
(1/x)*S(x)=∑(n=1->∞) 1/nx^n
=∑(n=1->∞) (1/n)*(1/x)^n
=-ln(1-1/x)
=ln[x/(x-1)]
S(x)=x*ln[x/(x-1)]
收敛域为x<=-1或x>1