已知数列an的前n项和sn=1+(r-1)an(常数r不等于2) 若limSn=1 求r已知数列an的前n项和sn=1+(r-1)an(常数r不等于2) 若limSn=1 求r的取值范围
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![已知数列an的前n项和sn=1+(r-1)an(常数r不等于2) 若limSn=1 求r已知数列an的前n项和sn=1+(r-1)an(常数r不等于2) 若limSn=1 求r的取值范围](/uploads/image/z/5126055-15-5.jpg?t=%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97an%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8Csn%3D1%2B%28r-1%29an%28%E5%B8%B8%E6%95%B0r%E4%B8%8D%E7%AD%89%E4%BA%8E2%29+%E8%8B%A5limSn%3D1+%E6%B1%82r%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97an%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8Csn%3D1%2B%28r-1%29an%28%E5%B8%B8%E6%95%B0r%E4%B8%8D%E7%AD%89%E4%BA%8E2%29+%E8%8B%A5limSn%3D1+%E6%B1%82r%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83%E5%9B%B4)
已知数列an的前n项和sn=1+(r-1)an(常数r不等于2) 若limSn=1 求r已知数列an的前n项和sn=1+(r-1)an(常数r不等于2) 若limSn=1 求r的取值范围
已知数列an的前n项和sn=1+(r-1)an(常数r不等于2) 若limSn=1 求r
已知数列an的前n项和sn=1+(r-1)an(常数r不等于2) 若limSn=1 求r的取值范围
已知数列an的前n项和sn=1+(r-1)an(常数r不等于2) 若limSn=1 求r已知数列an的前n项和sn=1+(r-1)an(常数r不等于2) 若limSn=1 求r的取值范围
a(1) = s(1) = 1 + (r-1)a(1),a(1) = 1/(2-r).
s(n) = 1 + (r-1)a(n),
r = 1时,s(n) = 1,a(1) = 1,a(n) = 1.
lim_{n->无穷}s(n) = 1,满足题意.
r不为1时,
s(n+1) = 1 + (r-1)a(n+1),
a(n+1) = s(n+1) - s(n) = (r-1)a(n+1) - (r-1)a(n),
a(n+1) = [(1-r)/(2-r)]a(n),
{a(n)}是首项为a(1)=1/(2-r),公比为(1-r)/(2-r)的等比数列.
a(n) = [1/(2-r)][(1-r)/(2-r)]^(n-1),
s(n) = 1 + (1-r)a(n) = 1 + [(1-r)/(2-r)]^n,
0 = lim_{n->无穷}[(1-r)/(2-r)]^n,
|(1-r)/(2-r)| < 1,
|1-r| < |2-r|,
1 - 2r + r^2 < 4 - 4r + r^2,
2r < 3,
r < 3/2.
综合,有,
r的取值范围为,r< 3/2