am - an = (m-n) d = 2/n - 2/m = 2(m-n)/mn
d = 2/mn
S19 = (am +an +d)/2 *19
= (2/n + 2/m + 2/mn)/2 *19
= (1/n+1/m + 1/mn) *19
= ((m+n)/mn + 1/mn) *19
= 20 *19/mn
mn取最大值时 S19有最小值
mn = (19-n)n = 19n- n^2
f(x) = -x^2 +19x 开口向下,对称轴x=19/2
x < 19/2时 单调增,x> 19/2时单调减.
所以
[1,19/2)中,整数n,f(n)最大值= f(9) = 90
(19/2,正无穷)区间上,整数n,f(n)最大值=f(10) = 90
所以S19最小值
= 20*19/90 = 38/9