Chứng minh: S = 1 + 1/2 + 1/3 + ... + 1/64 >4

Đề bài: Chứng minh:  $S=1+ \frac{1}{2}+\frac{1}{3}+...+\frac{1}{63}+\frac{1}{64}>4$

Trả lời

Hướng dẫn giải:

Ta có:

$$\begin{gathered} S=1+\frac{1}{2}+\frac{1}{3}+\cdot \cdot \cdot +\frac{1}{63}+\frac{1}{64} \\ =1+\frac{1}{2}+\left(\frac{1}{3}+\frac{1}{4}\right)+\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)+\cdot \cdot \cdot +\left(\frac{1}{33}+\frac{1}{34}+\cdot \cdot \cdot +\frac{1}{64}\right) \\ >1+\frac{1}{2}+\left(\frac{1}{4}+\frac{1}{4}\right)+\left(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}\right)+\cdot \cdot \cdot +\left(\frac{1}{64}+\cdot \cdot \cdot +\frac{1}{64}\right) \\ =1+\frac{1}{2}+2\cdot \frac{1}{4}+4\cdot \frac{1}{8}+8\cdot \frac{1}{16}+16\cdot \frac{1}{32}+32\cdot \frac{1}{64} \\ =1+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}=4 \end{gathered}$$Vậy $S=1+\frac{1}{2}+\frac{1}{3}+\cdot \cdot \cdot +\frac{1}{63}+\frac{1}{64}>4$