Câu 1 : Tính

S=\(\frac{10}{7}\)+\(\frac{10}{7^2}\)+\(\frac{10}{7^3}\)+...+\(\frac{10}{7^{10}}\)

Câu 2:Cho A=\(\frac{4}{15\cdot19}\)+\(\frac{4}{19\cdot23}\)+....+\(\frac{4}{399\cdot403}\).CMR:

\(\frac{16}{81}\)<A<\(\frac{16}{80}\)

Bài 1:CMR:\(\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+...+\frac{1}{15}<2\)

Bài 2: \(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{1}{x.\left(x+1\right)}=\frac{99}{101}\)

Bài 3:\(A=\frac{8}{9}.\frac{15}{16}.\frac{24}{25}.....\frac{2449}{2500}\)

Bài 4:CMR:\(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{49}-\frac{1}{50}=\frac{1}{26}+\frac{1}{27}+\frac{1}{28}+...+\frac{1}{50}\)

a) \(Cho\)\(S=\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+.....+\frac{1}{60}\)

\(Chứng\) \(minh\) \(\frac{3}{5}< S< \frac{4}{5}\)

b) \(Chứng\)\(minh\)\(\frac{1}{41}+\frac{1}{42}+\frac{1}{43}+.....+\frac{1}{100}>\frac{7}{10}\)

c) \(Chứng\)\(minh\)\(\frac{3}{10}+\frac{3}{11}+\frac{3}{12}+\frac{3}{13}+\frac{3}{14}\) không là số tự nhiên

d) \(Chứng\)\(minh\)\(\frac{1}{5}< D< \frac{1}{10}\) \(với\)\(D=\frac{1}{2}.\frac{3}{4}.\frac{5}{6}.....\frac{99}{100}\)

dạng 1 : so sánh 

a) P = \(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{2013^2}+\frac{1}{2014^2}\)và Q = \(1\frac{3}{4}\)

dạng 2 : toán chứng minh 

1. cho S = \(\frac{1}{101}+\frac{1}{102}+...+\frac{1}{130}\)chứng minh rằng : \(\frac{1}{4}< S< \frac{91}{330}\)

2. cho S = \(\frac{5}{20}+\frac{5}{21}+\frac{5}{22}+...+\frac{5}{49}\). CMR : 3 < S < 8

3. CMR : \(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2^{1999}}>1000\)