Chứng tỏ rằng:

\(\frac{1}{5}+\frac{1}{13}+\frac{1}{25}+....+\frac{1}{10^2+11^2}<\frac{9}{20}\)

Chứng minh \(\forall n\in\)N* thì\(\frac{1}{5}+\frac{1}{13}+\frac{1}{25}+...+\frac{1}{n^2+\left(n+1\right)^2}< \frac{9}{20}\)

1.CMR:

a) Cho a, b, c là các số nguyên dương

\(1<\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2\)

b) \(S3=\frac{1}{5}+\frac{1}{13}+\frac{1}{25}+...+\frac{1}{10^2+11^2}<\frac{9}{20}\)

Chứng minh rằng:

1)B=\(\frac{4}{3}+\frac{10}{9}+\frac{28}{27}+...+\frac{3^{98}+1}{3^{98}}< 100\)

2)C=\(\frac{5}{5.8.11}+\frac{5}{8.11.14}+...+\frac{5}{302.305.308}\)<\(\frac{1}{48}\)

3)D=\(\frac{11}{9}+\frac{18}{16}+\frac{27}{25}+...+\frac{1766}{1764}\)

\(40\frac{20}{43}< D< 40\frac{20}{21}\)

Bài 1 ; \(A=\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+......+\frac{1}{1+2+3+4+.....+2010}\)

Bài 2 : CHỨNG MINH RẰNG: Với mọi số nguyên n>1 , ta có :

\(\frac{1}{5}+\frac{1}{13}+\frac{1}{25}+.....+\frac{1}{n^2+\left(n+1\right)^2}< \frac{9}{20}\)

Chứng minh :

1,C=\(\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+...+\frac{1}{70}.C< \frac{3}{4}\)

2,D=\(\frac{1}{5^2}+\frac{1}{9^2}+...+\frac{1}{409^2}< \frac{1}{12}\)

3,E=\(\frac{5}{5.8.11}+\frac{5}{8.11.14}+...+\frac{5}{302.305.308}< \frac{1}{48}\)

Bài 1 : Cho A = \(\frac{1}{2}.\frac{3}{4}.\frac{5}{6}.\frac{7}{8}...\frac{79}{80}\)

Chứng minh rằng A < \(\frac{1}{9}\)

Bài 4 : Chứng minh rằng:  1.3.5.7....19 = \(\frac{11}{2}.\frac{12}{2}.\frac{13}{2}...\frac{20}{2}\)

Chứng minh:

c.\(\frac{11}{15}< \frac{1}{21}+\frac{1}{22}+\frac{1}{23}+...+\frac{1}{59}+\frac{1}{60}< \frac{3}{2}\)

b.\(\frac{1}{5}+\frac{1}{13}+\frac{1}{25}+\frac{1}{41}+\frac{1}{61}+\frac{1}{85}+\frac{1}{113}< \frac{1}{2}\)

a.\(\frac{1}{4}+\frac{1}{16}+\frac{1}{36}+\frac{1}{64}+\frac{1}{100}+\frac{1}{144}+\frac{1}{196}< \frac{1}{2}\)

Chứng minh rằng:

\(\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+...+\frac{1}{20}<\frac{1}{2}\)