必要性:
∑f(1/n)绝对收敛,则limf(1/n)=0,n->∞
∴f(0)=0 =>f'(0)=limnf(1/n),若f'(0)≠0
记an=f'(0)/n,则有lim|f(1/n)|/|an|=1
∴∑|f(1/n)|的敛散性和∑|an|相同
而∑|an|=f'(0)|∑1/n是发散的,∴∑|f(1/n)|也发散,矛盾
∴f'(0)=0
充分性:
∵f''(0)存在,∴f(x)在x=0的某个邻域内有一阶导数
∴limf(x)/x²=limf'(x)/(2x)=(1/2)lim(f’(x)-f'(0))/(x-0)=f''(0)/2,x->0
∴limn²f(1/n)=f''(0)/2,n->∞,即limn²|f(1/n)|=|f''(0)|/2
即∑|f(1/n)|的敛散性与∑|f''(0)|/(2n²)相同
而∑|f''(0)|/(2n²)=|f''(0)/2|∑1/n²是收敛的
∴∑f(1/n)绝对收敛