Question

用数学归纳法证明1/n+1/(n+1)+1/(n+2)+……1/n²>1(n∈N*,n>1)

Answer 1

n=2略

n=k时有1/k+1/(k+1)+……+1/k²>1

k≥2

令a=1/k+1/(k+1)+……+1/k²>1

则n=k+1

1/(k+1)+1/(k+2)+……+1/(k+1)²

=a-1/k+1/(k²+1)+……+1/(k+1)²

因为1/(k²+1)>1/(k+1)²

1/(k²+2)>1/(k+1)²

……

所以a-1/k+1/(k²+1)+……+1/(k+1)²>a-1/k+1/(k+1)²+……+1/(k+1)²

=a-1/k+(2k+1)*1/(k+1)²

=a+(2k²+k-k²-2k-1)/k(k+1)²

=a+[(k-1/2)²-5/4]k(k+1)²

k≥2

所以a+[(k-1/2)²-5/4]k(k+1)²>a>1

所以n=k+1

1/(k+1)+1/(k+2)+……+1/(k+1)²>1

综上,……