Question

CMR:

\(A=\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+...+\frac{1}{2019\times2020}< 1\)

\(B=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{50^2}< \frac{3}{4}\)

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

Answer 1

\(A=\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{2019.2020}\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{2019}-\frac{1}{2020}\)

\(=1-\frac{1}{2020}< 1\)

Vậy \(A< 1\left(đpcm\right)\)

Answer 2

\(B=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{50^2}< \frac{1}{4}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{49.50}\)

\(\Leftrightarrow B< \frac{1}{4}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{49}-\frac{1}{50}\)

\(\Leftrightarrow B< \frac{1}{4}+\frac{1}{2}-\frac{1}{50}\)

\(\Leftrightarrow B< \frac{1}{4}+\frac{1}{2}\)

\(\Leftrightarrow B< \frac{3}{4}\left(đpcm\right)\)

Answer 3

\(C=\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}...+\frac{1}{100!}< 1+\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{99.100}\)

\(\Leftrightarrow C< 1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{99}-\frac{1}{100}\)

\(\Leftrightarrow C< 2-\frac{1}{100}\)

\(\Leftrightarrow C< 2\left(đpcm\right)\)