Question

已知A=1+x+2x^2+3x^3+…… nx^n,若x=1/2,则lim(A+n/2^n)=

Answer 1

∵A=1+x+2x²+3x³+.+nx^n

==>xA=x+x²+2x³+3x^4+.+nx^(n+1)

==>(1-x)A=1+x²+x³+.+x^n-nx^(n+1)=(1-x^(n+1))/(1-x)-x-nx^(n+1)

∴A=(1-x^(n+1))/(1-x)²-(x+nx^(n+1))/(1-x)

∵x=1/2

∴A=(1-(1/2)^(n+1))/(1-1/2)²-(1/2+n(1/2)^(n+1))/(1-1/2)

=3-1/2^(n-1)-n/2^n

==>A+n/2^n=3-1/2^(n-1)

故lim(n->∞)(A+n/2^n)=3-0=3.