Question

已知数列an=1/n（n+1）（n+2） 求Sn

Answer 1

解：an=1/[n(n+1)(n+2)]，
=[1/(n+1)]* [ 1/[n(n+2)]
=[1/(n+1)]* {1/2 *[1/n-1/(n+2)]}
=1/2 *[1/n(n+1)-1/(n+1)(n+2)]
=1/[2n(n+1)]-1/[2(n+1)(n+2)].
Sn=a1+a2+……+a(n-1)+an
={1/[2*1(1+1)] -1/[2(1+1)(1+2)]} +{1/[2*2(2+1)] -1/[2(2+1)(2+2)]} +
……+{1/[2(n-1)n] -1/[2n(n+1)]}+{1/[2n(n+1)] -1/[2(n+1)(n+2)]}
=1/4-1/[2(n+1)(n+2)].
即Sn=1/4-1/[2(n+1)(n+2)].

【另，显然，当n趋于无穷大时，Sn取得极限=1/4】