Question

Cho tổng S=1/31 1/32 .... 1/60 . Cmr S<4/5    

Answer 1

ý bạn là: \(\frac{1}{31}+\frac{1}{32}+...+\frac{1}{60}\left(CMR:S< \frac{4}{5}\right)\)

Tổng S có 30 số hạng. Vậy tách S ra thành 3 nhóm, mỗi nhóm có 10 số.

\(S=\left(\frac{1}{31}+\frac{1}{32}+...+\frac{1}{40}\right)+\left(\frac{1}{41}+\frac{1}{42}+...+\frac{1}{50}\right)+\left(\frac{1}{51}+\frac{1}{52}+...+\frac{1}{60}\right)\)

\(\Leftrightarrow S< \left(\frac{1}{30}+\frac{1}{30}+...+\frac{1}{30}\right)+\left(\frac{1}{40}+\frac{1}{40}+....+\frac{1}{40}\right)+\left(\frac{1}{50}+\frac{1}{50}+...+\frac{1}{50}\right)\)

\(\Leftrightarrow S< \frac{1}{3}+\frac{1}{4}+\frac{1}{5}=\frac{20}{60}+\frac{15}{60}+\frac{12}{60}=\frac{47}{60}< \frac{48}{60}=\frac{4}{5}\)