Question

是否存在常数a,b,c,使等式(1/n)3+(2/n)3+(3/n)+.+(n/n)3=(an2+bn+c)/n对一切n

Answer 1

存在.a=0.25,b=0.5,c=0.25

证：(1/n)3+(2/n)3+(3/n)+.+(n/n)3

=(1^3+2^3+3^3+.+(n)^3)/n^3

=[n(n+1)/2]^2/n^3

=(n+1)^2/(4n)

=(0.25n^2+0.5n+0.25)/n

=(an2+bn+c)/n