求极限lim(n→∞)[(2+3)/5+(2^2+3^2)/(5^2)+(2^3+3^3)/(5^3)+...+(2^n+3^n)/(5^n)]=?

Question

求极限lim(n→∞)[(2+3)/5+(2^2+3^2)/(5^2)+(2^3+3^3)/(5^3)+...+(2^n+3^n)/(5^n)]=?

Answer 1

lim(n→∞)[(2+3)/5+(2^2+3^2)/(5^2)+(2^3+3^3)/(5^3)+...+(2^n+3^n)/(5^n)]
=lim(n→∞)[(2/5)+(2/5)^2+…+(2/5)^n+(3/5)+(3/5)^2+...+(3/5)^n]
=lim(n→∞)(2/5)[1-(2/5)^n]/(1-2/5)+lim(n→∞)(3/5)[1-(3/5)^n]/(1-3/5)
=(2/5)/(3/5)+(3/5)/(2/5)
=2/3+3/2
=13/6