1) Cho S = 1/4+2/4^2 +3/4^3 +4/4^4+...+2023/4^3023 Chứng minh S<1/2

S = = 1/4+2/4^2 +3/4^3 +4/4^4+...+2023/4^2023
4S = 1 + $\frac{2}{4}$ + $\frac{3}{4^{2}}$ + ... + $\frac{2023}{4^{2022}}$

4S-S = 1 + $\frac{2}{4}$ + $\frac{3}{4^{2}}$ + ... + $\frac{2023}{4^{2022}}$ - ($\frac{1}{4}$ + $\frac{2}{4^{2}}$ + ... $\frac{2023}{4^{2022}}$)

3S = 1 + $\frac{1}{4}$ + $\frac{1}{4^{2}}$ + $\frac{1}{4^{3}}$ + ... + $\frac{1}{4^{2022}}$ - $\frac{2023}{4^{2022}}$

3S - S = 1 + $\frac{1}{4}$ + $\frac{1}{4^{2}}$ + $\frac{1}{4^{3}}$ + ... + $\frac{1}{4^{2022}}$ - $\frac{2023}{4^{2022}}$ - ($\frac{1}{4}$ + $\frac{2}{4^{2}}$ + ... $\frac{2023}{4^{2022}}$)

2S = 1 - $\frac{1}{4}$ - $\frac{1}{4^{2}}$ - $\frac{2}{4^{3}}$ - ... - $\frac{2021}{4^{2022}}$ < 1
=> S < $\frac{1}{2}$