tính tổng dãy số thì dễ nhưng hãy viết rõ ràng hơn

\(S=\dfrac{1}{1.2}+\dfrac{2}{1.2.3}+........+\dfrac{99}{1.2.......100}\)

\(=\dfrac{1}{2!}+\dfrac{2}{3!}+....+\dfrac{99}{100!}\)

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

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

\(=1-\dfrac{1}{100!}< 1\)

\(\Leftrightarrow S< 1\left(đpcm\right)\)

\(\frac{1}{1.2}+\frac{1}{1.2.3}+\frac{1}{1.2.3.4}+...+\frac{1}{1.2.3.4.....1000}\)

Có: \(\frac{1}{1.2.3.4}< \frac{1}{3.4}\)

\(\frac{1}{1.2.3.4.5}< \frac{1}{4.5}\)

..................................

\(\frac{1}{1.2.3.4.....1000}< \frac{1}{999.1000}\)

=>\(\frac{1}{1.2.3.4}+\frac{1}{1.2.3.4.5}+...+\frac{1}{1.2.3.4.....1000}< \frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{999.1000}\)

=> \(\frac{1}{1.2.3.4}+\frac{1}{1.2.3.4.5}+...+\frac{1}{1.2.3.4.....1000}< \frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{999}-\frac{1}{1000}\)

=> \(\frac{1}{1.2.3.4}+\frac{1}{1.2.3.4.5}+...+\frac{1}{1.2.3.4.....1000}< \frac{1}{3}-\frac{1}{1000}\)

=> \(\frac{1}{1.2}+\frac{1}{1.2.3}+\frac{1}{1.2.3.4}+...+\frac{1}{1.2.3.4.....1000}< \frac{1}{2}+\frac{1}{1.2.3}+\frac{1}{3}-\frac{1}{1000}\)

=> \(\frac{1}{1.2}+\frac{1}{1.2.3}+\frac{1}{1.2.3.4}+...+\frac{1}{1.2.3.4.....1000}< \frac{999}{1000}< \frac{1000}{1000}\)

=>\(\frac{1}{1.2}+\frac{1}{1.2.3}+\frac{1}{1.2.3.4}+...+\frac{1}{1.2.3.4.....1000}< 1\)

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

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

\(S=1-\frac{1}{100!}< 1\)

Vậy S<1

thánh đây rồi , đơn giản vậy em nghĩ mãi k ra , cảm ơn anh nhiều

B _^chăng:D

Đặt A = \(1+\frac{1}{1.2}+\frac{1}{1.2.3}+\frac{1}{1.2.3.4}+...+\frac{1}{1.2.3....n}\)

Ta có: \(\frac{1}{1.2}=\frac{1}{1.2}\)

\(\frac{1}{1.2.3}=\frac{1}{2.3}\)

\(\frac{1}{1.2.3.4}< \frac{1}{3.4}\)

..............

\(\frac{1}{1.2.3....n}< \frac{1}{\left(n-1\right)n}\)

Cộng vế với vế ta được:

\(A< 1+\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{\left(n-1\right)n}=1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{n-1}-\frac{1}{n}=1+1-\frac{1}{n}=2-\frac{1}{n}< 2\)(đpcm)