一道数列题(数列/凹函数/凹凸性/二阶导数/函数)

1个回答

  • a(n)=n,

    s(n)=n(n+1)/2.

    1/[s(n)]^2 = 4/[n(n+1)]^2,

    f(x)= 4/[x(x+1)]^2 ,x>=1,f(x) > 0.

    g(x)=ln[f(x)] = ln(4) - 2ln(x)-2ln(x+1)

    g'(x) = f'(x)/f(x) = -2/x - 2/(x+1),

    f'(x)=f(x)*g'(x) = 4/[x(x+1)]^2 *[-2/x - 2/(x+1)],

    g''(x)=2/x^2 + 2/(x+1)^2 >0,

    f''(x) = f'(x)*g'(x) + f(x)*g''(x) = f(x)*[g'(x)]^2 + f(x)*g''(x) = f(x){[g'(x)]^2 + g''(x)}>0

    f(x) 为凸函数.

    m>=1,n>=1时,

    [f(m)+f(n)]/2 >= f[(m+n)/2] = f(p),

    1/[s(n)]^2 + 1/[s(m)]^2 >= 2/[s(p)]^2.