Question

[Chàng Trai 2_k_7]

CMR:

a)\(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^7}\)  <1

b)\(\frac{1}{3}+\frac{2}{3^2}+\frac{3}{3^3}+\frac{4}{4^4}+....+\frac{101}{3^{101}}\),<3/4

nhanh nhé

Answer 1

[Ngoc Anhh]

a) Ta có 

\(A=\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^7}\)

\(2A=1+\frac{1}{2}+...+\frac{1}{2^6}\)

\(2A-A=\left(1+\frac{1}{2}+...+\frac{1}{2^6}\right)-\left(\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^7}\right)\)

\(A=1-\frac{1}{2^7}\)

Do \(1-\frac{1}{2^7}< 1\Rightarrow A< 1\left(đpcm\right)\)