Chứng minh rằng:

a)\(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2010^2}\)<1

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

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

d)\(\frac{1}{3^3}+\frac{1}{4^3}+\frac{1}{5^3}+...+\frac{1}{n^3}\)<\(\frac{1}{12}\)\(\left(n\in N;n\ge3\right)\)

e)\(\frac{3}{4}+\frac{5}{36}+\frac{7}{144}+...+\frac{2n+1}{n^2\left(n+1\right)^2}\)<1 (n nguyên dương)

g)\(\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+...+\frac{1}{2048}\)>3

h)\(\left(\frac{2}{1}\right)\left(\frac{4}{3}\right)\left(\frac{6}{5}\right)...\left(\frac{200}{199}\right)\)

1. Cho I =\(\frac{1}{4^2}+\)\(\frac{1}{6^2}+\)\(\frac{1}{8^2}+...+\frac{1}{2006^2}\)chứng minh: I < \(\frac{334}{2007}\)
2. Cho K = \(\frac{1}{5^2}+\frac{1}{9^2}+...+\frac{1}{409^2}\)chứng minh: K < \(\frac{1}{12}\)
3. cho M = \(\frac{3}{4}+\frac{8}{9}+\frac{15}{16}+...+\frac{2499}{2500}\)chứng minh: M > 48
4. cho N = \(\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+\frac{1}{1+2+3+4+5}+...+\frac{1}{1+2+3+4+...+59}\)chứng minh: N < \(\frac{2}{3}\)
5. cho S = \(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{37.38.39}\)chứng minh: S <\(\frac{1}{4}\)

1.chứng minh rằng A<\(\frac{1}{16}\) biết A=\(\frac{1}{5^2}+\frac{2}{5^3}+\frac{3}{5^4}+.....+\frac{99}{5^{100}}\)

2.tính (M-N)\(^3\) biết:

M=1-\(\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+.....+\frac{1}{2017}-\frac{1}{2018}+\frac{1}{2019}\)

N=\(\frac{1}{1010}+\frac{1}{1011}+.....+\frac{1}{2019}\)

 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 \(\frac{1}{3}-\frac{2}{3^2}+\frac{3}{3^3}-\frac{4}{3^4}+\frac{5}{3^5}-...+\frac{99}{3^{99}}-\frac{100}{3^{100}}< \frac{3}{16}\)

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

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