chứng minh :

a,A=1+\(\frac{1}{2^2}\)+\(\frac{1}{3^2}\)+\(\frac{1}{4^2}\)+...+\(\frac{1}{100^2}\)< 2

b,B=1+\(\frac{1}{2}\)+\(\frac{1}{3}\)+\(\frac{1}{4}\)+...+\(\frac{1}{63}\)< 6

c,C=\(\frac{1}{2}\).\(\frac{3}{4}\).\(\frac{5}{6}\).\(\frac{7}{8}\)...\(\frac{9999}{10000}\)< \(\frac{1}{100}\)

cm 

a,A=1+\(\frac{1}{2^2}\)+\(\frac{1}{3^2}\)+\(\frac{1}{4^2}\)+....+\(\frac{1}{100^2}\)<2

b,B=1+\(\frac{1}{2}\)+\(\frac{1}{3}\)+\(\frac{1}{4}\)+....+\(\frac{1}{63}\)<6

c,C=\(\frac{1}{2}\).\(\frac{3}{4}\).\(\frac{5}{6}\).\(\frac{7}{8}\)...\(\frac{9999}{10000}\)<\(\frac{1}{100}\)

Chứng minh rằng :

a)\(\frac{1}{2}\)-\(\frac{1}{4}\)+\(\frac{1}{8}\)-\(\frac{1}{16}\)+\(\frac{1}{32}\)-\(\frac{1}{64}\)<\(\frac{1}{3}\)

b)\(\frac{1}{3}\)-\(\frac{2}{3^2}\)+\(\frac{3}{3^3}\)-\(\frac{4}{3^4}\)+......+\(\frac{99}{3^{99}}\)-\(\frac{100}{3^{100}}\)<\(\frac{3}{16}\)

1,Tính

\(\frac{5^2}{6}\)+\(\frac{5^2}{66}\)+...+\(\frac{5^2}{806}\)

2,Tìm x

a, (\(\frac{2}{11.13}\)+\(\frac{2}{13.15}\)+...+\(\frac{2}{19.21}\)) - x +\(\frac{221}{231}\)=\(\frac{4}{3}\)

b, \(\frac{1}{3.4}\)+\(\frac{1}{4.5}\)+\(\frac{1}{5.6}\)+\(\frac{1}{6.7}\)+...+\(\frac{1}{x.\left(x+1\right)}\)=\(\frac{3}{10}\)

3, Chứng minh dãy số nhỏ hơn 1

\(\frac{1}{2^2}\)+\(\frac{1}{3^2}\)+\(\frac{1}{4^2}\)+...+\(\frac{1}{100^2}\)<1

Tính : 

a) A=2\(\frac{3}{7}\):(-12\(\frac{1}{2}\)+  15\(\frac{1}{7}\)) . (-3\(\frac{3}{10}\)) - (-\(\frac{34}{37}\)) . (2\(\frac{1}{10}\)-4\(\frac{2}{5}\))

b)B=3\(\frac{1}{10}\):(\(\frac{1}{2}\)-\(\frac{2}{5}\)) - 2\(\frac{1}{28}\): (\(\frac{2}{7}\)- \(\frac{3}{4}\))

Bai 1: Cho \(A\) = \(\left(\frac{1}{2^2}-1\right)\)\(.\)\(\left(\frac{1}{3^2}-1\right)\)\(.\)\(\left(\frac{1}{4^2}-1\right)\)\(...\)\(\left(\frac{1}{100^2}-1\right)\)\(.\)So sánh  \(A\)với \(\frac{-1}{2}\)

Bai 2;Chứng minh rằng;

           a) \(\frac{3}{1^2.2^2}\)+   \(\frac{5}{2^2.3^2}\)+   \(\frac{7}{3^2.4^2}\)+ ... +   \(\frac{19}{9^2.10^2}\)\(< \)\(1\)

           b) \(\frac{1}{3}\)+  \(\frac{2}{3^2}\)+\(\frac{3}{3^3}\)+ ... +  \(\frac{99}{3^{99}}\)+   \(\frac{100}{3^{100}}\)\(< \frac{3}{4}\)

1. Cho A=\(\frac{1}{1.101}\)+\(\frac{1}{2.102}\)+...+\(\frac{1}{10.110}\); B= \(\frac{1}{1.11}\)+\(\frac{1}{2.12}\)+...+\(\frac{1}{100.110}\)

Tính A : B

2. Cho A= 1+\(\frac{3}{2^3}\)+\(\frac{4}{2^4}\)+...+\(\frac{100}{2^{100}}\).Rút gọn A

a) A =(1-\(\frac{1}{2}\))x( 1-\(\frac{1}{3}\))x(1-\(\frac{1}{4}\))x ..... x(1-\(\frac{1}{2017}\))

b) B =5\(\frac{9}{10}\):\(\frac{3}{2}\)-(2\(\frac{1}{3}\)x 4\(\frac{1}{2}\)-2x2\(\frac{1}{3}\)):\(\frac{7}{4}\)

bài 1:thực hiện phép tính

a,\(\frac{65}{91}\)+\(\frac{-33}{55}\) b,\(\frac{36}{-84}\)+\(\frac{100}{450}\) c,\(\frac{3}{7}\)+\(\frac{1}{2}\)-\(\frac{-3}{70}\) d,\(\frac{3}{5}\)+\(\frac{-1}{25}\)-\(\frac{-35}{100}\)

bài 2:tìm x,bt

a,x-\(\frac{-1}{5}\)=\(\frac{7}{25}\) b,x+\(\frac{5}{11}\)=\(\frac{4}{-9}\) c,\(\frac{5}{9}\)+\(\frac{x}{-1}\)=\(\frac{-1}{3}\) d,|x-4|=|\(\frac{-6}{7}\)|+\(\frac{3}{2}\)

bài 3:tính nhanh(hợp lí)

a,A=\(\frac{2}{15}\)+\(\frac{2}{35}\)+\(\frac{2}{63}\)+\(\frac{2}{99}\)+\(\frac{2}{143}\)

b,B=\(\frac{1}{21}\)+\(\frac{1}{77}\)+\(\frac{1}{165}\)+\(\frac{1}{285}\)

c,so sánh C=\(\frac{1}{11}\)+\(\frac{1}{12}\)+\(\frac{2}{12}\)+.................+\(\frac{1}{20}\)với \(\frac{1}{2}\)

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