Question

高中数列第五题

高中数列第五题过程要有，谢谢

Answer 1

1²+(1+ 1/n)²+(1+ 2/n)²+...+[1+ (n-1)/n]²

=1+1+ 2/n +1/n²+1+4/n+4/n²+...+1+ 2(n-1)/n +(n-1)²/n²

=n+[2+4+...+2(n-1)]/n+ [1/n²+2²/n²+...+ (n-1)²/n²]

=n+2[1+2+...+(n-1)]/n +[1²+2²+...+(n-1)²]/n²

=n+2[n(n-1)/2]/n + (n-1)n[2(n-1)+1]/(6n²)

=n+ n-1 + (n-1)(2n-1)/(6n)

=2n-1 +(2n²-3n+1)/(6n)

=2n -1 +⅓n -½ +1/(6n)

=(7/3)n -(3/2) +1/(6n)

(1/n)[1²+(1+ 1/n)²+(1+ 2/n)²+...+[1+ (n-1)/n]²]

=(1/n)[(7/3)n -(3/2) +1/(6n)]

=(7/3) -(3/2)/n +1/(6n²)

n→∞，(3/2)/n→0，1/(6n²)→0

(7/3) -(3/2)/n +1/(6n²)→ 7/3

lim(1/n)[1²+(1+ 1/n)²+(1+ 2/n)²+...+[1+ (n-1)/n]²] =7/3

n→∞

选C

Answer 2