Cho tổng gồm 2014 số hạng S=1/4+2/4^2+3/4^3+….+2014/4^2014 Chứng minh S<1/2 03/09/2021 Bởi Genesis Cho tổng gồm 2014 số hạng S=1/4+2/4^2+3/4^3+….+2014/4^2014 Chứng minh S<1/2
4S=1+2/4+3/4^2+….+2014/4^2013 4S-S= (1+2/4+3/4^2+….+2014/4^2013) – (1/4+2/4^2+3/4^3+….+2014/4^2014) 3S=1+1/4+1/4^2+…..+1/4^2013-2014/4^2014 Đặt A= 1+1/4+1/4^2+…..+1/4^2013 ⇒4A=4+1+1/4+….+1/4^2012 4A-A=4-1/4^2013⇒A=4/3-1/3*4^2013 ⇒3S=4/3-1/3*4^2013-2014/4^2014⇒S=4/9-1/9*4^2013-2014/3*4^2014 Vì 4/9<1/2 nên 4/9-1/9*4^2013-2014/3*4^2014<1/2⇒S<1/2 Bình luận
4S=1+2/4+3/4^2+….+2014/4^2013
4S-S= (1+2/4+3/4^2+….+2014/4^2013) – (1/4+2/4^2+3/4^3+….+2014/4^2014)
3S=1+1/4+1/4^2+…..+1/4^2013-2014/4^2014
Đặt A= 1+1/4+1/4^2+…..+1/4^2013
⇒4A=4+1+1/4+….+1/4^2012
4A-A=4-1/4^2013⇒A=4/3-1/3*4^2013
⇒3S=4/3-1/3*4^2013-2014/4^2014⇒S=4/9-1/9*4^2013-2014/3*4^2014
Vì 4/9<1/2 nên 4/9-1/9*4^2013-2014/3*4^2014<1/2⇒S<1/2