C1: Cho A=\(\frac{1}{2}\)- \(\frac{2}{2^2}\)+ \(\frac{3}{2^3}\)- ... + \(\frac{99}{2^{99}}\)- \(\frac{100}{2^{100}}\) . Chứng minh rằng: A<\(\frac{2}{9}\)

C2: Cho B=\(\frac{1}{2^2}\)+ \(\frac{1}{4^2}\)+ \(\frac{1}{6^2}\)+ ... + \(\frac{1}{2000^2}\). Chứng minh rằng: B<\(\frac{1}{2}\)

   Mình vội lắm, ai làm xong đầu tiên mình TICK cho

 Chứng minh rằng :

a) S=\(\frac{3}{1.4}\)+\(\frac{3}{4.7}\)+...+\(\frac{3}{n(n+3)}\)< 1 .

b) Cho : S=\(\frac{1}{2^2}\)+\(\frac{1}{3^2}\)+\(\frac{1}{4^2}\)+...+\(\frac{1}{9^2}\). Chứng minh :\(\frac{2}{5}\)< S <\(\frac{8}{9}\).

   Gợi ý : Chứng minh \(\frac{1}{b}\)-\(\frac{1}{b+1}\)<\(\frac{1}{b^2}\)<\(\frac{1}{b-1}\)-\(\frac{1}{b}\) ( b\(\varepsilon\)\(ℕ\), b > 1 ) .

Bài 1:

a, Cho S=\(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{9^2}\) .Chứng minh rằng \(\frac{2}{5}< S< \frac{8}{9}\)

b, Tìm x thuộc z để phân số \(\frac{x^2-5x-1}{x+2}\)có giá trị là số nguyên

c, Chứng minh rằng \(\left(\frac{7}{65}+1\right)\left(\frac{7}{84}+1\right)\left(\frac{7}{105}+1\right)\left(\frac{7}{124}+1\right)...\left(\frac{7}{153+1}\right)\left(\frac{7}{560}+1\right)< 2\)

d, Chứng minh rằng \(\frac{1}{3}-\frac{2}{3^2}+\frac{3}{3^3}-\frac{4}{3^4}+\frac{5}{3^5}-...+\frac{99}{3^{99}}-\frac{100}{3^{100}}< \frac{3}{16}\)

1)Chứng minh các phân số sau là các phân số tối giản:

a)\(A=\frac{12n+1}{30n+2}\)

b)\(B=\frac{14n+17}{21n+25}\)

2)Chứng minh rằng:

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

Bai 1; a) Cho \(B\) = \(\frac{1}{2}\)+   (\(\frac{1}{2}\))²  +     (\(\frac{1}{2}\))³  +   (\(\frac{1}{2}\))⁴  + ... +  (\(\frac{1}{2}\))⁹⁸  +   (\(\frac{1}{2}\))⁹⁹ . Chứng minh rằng  \(B\) \(< 1\)

          b) Cho \(C\) = \(\frac{1}{3}\)+    \(\frac{1}{3^2}\)+   \(\frac{1}{3^3}\)+...+   \(\frac{1}{3^{98}}\)+   \(\frac{1}{3^{99}}\).  Chứng minh rằng \(C\) \(< \)\(\frac{1}{2}\)

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

Bài 1:So sánh Avà B biết rằng:

A=\(\frac{10^{15}+1}{10^{16}+1};\) B=\(\frac{10^{16}+1}{10^{17}+1}\)

A=\(\frac{3}{8^3}+\frac{7}{8^4}\); B=\(\frac{7}{8^3}+\frac{3}{8^4}\)

A=\(\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+.......+\frac{1}{19}+\frac{1}{20};\) B=\(\frac{1}{2}\)

Bài 2:Dạng tính tổng đặc biệt:

\(A=\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+.....+\frac{1}{99\cdot100}\)

\(B=\frac{2}{1\cdot3}+\frac{2}{3\cdot5}+\frac{2}{5\cdot7}+.....+\frac{2}{99\cdot101}\)

\(C=\frac{3^2}{10}+\frac{3^2}{40}+\frac{3^2}{88}+......+\frac{3^2}{340}\)

\(D=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+......+\frac{1}{3^8}\)

\(E=\left(1-\frac{1}{2}\right)\cdot\left(1-\frac{1}{3}\right)\cdot\left(1-\frac{1}{4}\right).......\left(1-\frac{1}{99}\right)\)

Bài 3:Dạng chứng minh:

\(A=1+\frac{1}{2}+\frac{1}{3}+......+\frac{1}{99}.\)Chứng minh rằng A chia hết cho 100

A=\(\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+...+\frac{1}{70}\).Chứng minh rằng A>\(\frac{4}{3}\)

Cho M = \(\frac{1}{2}\)× \(\frac{3}{4}\)×\(\frac{5}{6}\)× ...×\(\frac{99}{100}\)

Cho N = \(\frac{2}{3}\)× \(\frac{4}{5}\)×\(\frac{6}{7}\)× ...×\(\frac{100}{101}\)

Chứng minh rằng M < N

Tính M . N

Chứng minh rằng M < \(\frac{1}{10}\)

chứng minh rằng

B=\(\frac{1}{2}\)*\(\frac{3}{4}\)*\(\frac{5}{6}\)*...*\(\frac{199}{200}\).CHỨNG MINH B²<\(\frac{1}{201}\)

C= 1+\(\frac{1}{2}\)+\(\frac{1}{3}\)+\(\frac{1}{4}\)+...+\(\frac{1}{2^{100}-1}\)CHỨNG MINH C<100

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