$\displaystyle{\frac{1}{1+a+ab} + \frac{1}{1+b+bc} + \frac{1}{1+c+cd} + \frac{1}{1+d+da} > 1}$
2) Cho a,b,c là các số thực dương. Chứng minh rằng:
$\displaystyle{1<\frac{a^2}{a^2+bc} + \frac{b^2}{b^2+ca} + \frac{c^2}{c^2+ab}<2 }$ (không dùng bunhia)
3) Chứng minh rằng với mọi số nguyên dương n ta có:
a)$1<\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+...+\frac{1}{3n+1}<2$
b)$\frac{1}{5}+\frac{1}{13}+\frac{1}{25}...+\frac{1}{n^2+(n+1)^2}<\frac{1}{2}$
c)$\frac{3}{4}+\frac{5}{36}+\frac{7}{144}+...+\frac{2n+1}{n^2(n+1)^2}<1$
d)$\frac{1}{7}+\frac{1}{13}+\frac{1}{21}+...+\frac{1}{n^2+3(n+1)}<\frac{n}{2(n+2)}$
4)Chứng minh rằng:
a) $\frac{4}{3}<\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+...+\frac{1}{70}<\frac{5}{2}$
b) $\frac{1}{15}<\frac{1}{2}.\frac{3}{4}.\frac{5}{6}...\frac{99}{100}<\frac{1}{10}$
5)Chứng minh rằng với mọi số tự nhên $n>1$:
a) $2\sqrt{n}-3<\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+...+\frac{1}{\sqrt{n}}<2\sqrt{n}-2 $
b)$\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+...+\frac{n-1}{n!}<1 $
c)$\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{n^2}>1$
d)$\frac{1}{3}+\frac{2}{2^2}+\frac{3}{3^3}+...+\frac{n}{3^n}<\frac{3}{4}$
e)$0,71<\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+...+\frac{1}{(n-1)!}+\frac{1}{(n+1)!}<0,72$
f)$\frac{2^3-1}{2^3+1}.\frac{3^3-1}{3^3+1}...\frac{n^3-1}{n^3+1}>\frac{2}{3}$
g)$\frac{1}{1^4+4}+\frac{3}{3^4+4}+\frac{5}{5^4+4}+...+\frac{2n-1}{(2n-1)^4+4}<\frac{1}{4}$
h)$\frac{1}{2\sqrt{n}}<\frac{1}{2}.\frac{3}{4}.\frac{5}{6}...\frac{2n-1}{2n}$
Bài viết đã được chỉnh sửa nội dung bởi Yagami Raito: 07-03-2014 - 18:23
