S=(1+1/2)+(1+2/2^2)+(1+3/2^3)+...+(1+2014/2^2014)
CM :S<2016
Cần c/m: $A= \dfrac 3{2^3} + \dfrac 4{2^4} ... + \dfrac{2014}{2^{2014}} < 1$.
Đặt $\dfrac 12 = a \Rightarrow A = 3a^3 + 4a^4 + ... + 2014a^{2014}$
$2A = 3a^2.2a + 4a^3.2a + ... + 2014a^{2013}.2a = 3a^2 + 4a^3 + ... + 2014a^{2013}$
$\Rightarrow A = 2A - A = 2a^2 + a^2 + a^3 + ... + a^{2013} - 2014a^{2014}$
$= 2a^2 + \dfrac{a^2(1 + a + ... + a^{2011})(1 - a)}{1 - a} - 2014a^{2014}$
$= 2a^2 + \dfrac{a^2 (1 - a^{2012})}{1 - a} - 2014a^{2014}$
$= \dfrac 2{2^2} + \dfrac12 (1 - \dfrac1{2^{2012}}) - \dfrac{2014}{2^{2014}} = 1 - \dfrac1{2^{2013}} - \dfrac{2014}{2^{2014}} < 1$ (đpcm)
Last edited:
[TẶNG BẠN] TRỌN BỘ Bí kíp học tốt 08 môn
