Question

[Kỉ niệm tuổi thơ]

Chứng minh  (1/2!+2/3!+3/4!+4/5!+5/6!+...+99/100!) <1

Answer 1

[Kira Kira]

\(\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+...+\frac{99}{100!}\)

= \(\frac{2-1}{2!}+\frac{3-1}{3!}+\frac{4-1}{4!}+....+\frac{100-1}{100!}\)

= \(1-\frac{1}{2!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{3!}-\frac{1}{4!}+....+\frac{1}{99!}-\frac{1}{100!}\)

= \(1-\frac{1}{100!}<1\)

=> \(\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+...+\frac{99}{100!}<1\)(Đpcm)