Chứng minh rằng:
\(\frac{2}{3^3}+\frac{3}{3^3}+\frac{4}{3^4}+...+\frac{2016}{3^{2016}}< \frac{5}{12}\)
Chứng minh rằng : \(B=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+....+\frac{1}{2^{2016}-2}+\frac{1}{2^{2016}-1}>1008\)
A = \(\frac{\frac{3}{4}-\frac{3}{11}+\frac{3}{13}}{\frac{5}{4}-\frac{5}{11}+\frac{5}{13}}+\frac{\frac{1}{2}-\frac{1}{3}+\frac{1}{4}}{\frac{5}{4}-\frac{5}{6}+\frac{5}{8}}\)
B = \(\frac{2^{12}.3^5-4^6.9^2}{\left(2^2.3\right)^6+8^4.3^5}-\frac{5^{10}.7^3-25^5.49}{\left(125.7\right)^3+5^9.14^3}\)
C = \(\frac{\left(a^{2016}+b^{2016}\right)^{2017}}{\left(c^{2016}+d^{2016}\right)^{2017}}\)= \(\frac{\left(a^{2017}-b^{2017}\right)^{2016}}{\left(c^{2017}-d^{2017}\right)^{2016}}\)
tinh B=\(\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...+\frac{1}{2016}}{\frac{2016}{1}+\frac{2003}{2}+\frac{2002}{3}+...+\frac{1}{2016}}\)
chứng minh rằng:\(\frac{1}{5^3}+\frac{1}{6^3}+....+\frac{1}{2016^3}+\frac{1}{2017^3}< \frac{1}{40}\)
Cho \(M=\frac{3}{4}+\frac{3^2}{4^2}+\frac{3^3}{4^3}+...+\frac{3^{2016}}{4^{2016}}\) Tìm phần nguyên của M
Câu 1:Rút gọn các biểu thức:
A=\(\frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+...+\frac{1}{97.99}-\frac{5}{4}.\frac{13}{99}+\frac{5}{99}.\frac{1}{4}\)
Câu 2: So sánh:
A=\(\frac{2013}{2014}+\frac{2016}{2015}\)và \(\frac{2014}{2015}+\frac{2017}{2016}\)
Câu 3: Cho f(x)=ax2+bx+c. Biết 7a+b=0. Chứng minh rằng: f(10).f(-3)\(\ge\)0
Bài 1 : Tính :
a)\(\frac{\left(13\frac{1}{4}-2\frac{5}{27}-10\frac{5}{6}\right)\times230\frac{1}{5}+46\frac{3}{4}}{\left(1\frac{3}{10}+\frac{10}{3}\right)\div\left(12\frac{1}{3}-14\frac{2}{7}\right)}\)
b) \(\frac{2^{12}\times3^5-4^6\times9^2}{\left(2^4\times3\right)^6+8^4\times3^5}-\frac{5^{10}\times7^3-25^5\times49^2}{\left(125\times7\right)^3+5^9\times14^3}\)
c)P=\(\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+....+\frac{1}{2016}}{\frac{2015}{1}+\frac{2014}{2}+\frac{2013}{3}+....+\frac{1}{2015}}\)
Chứng minh: \(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{2015}}+\frac{1}{3^{2016}}<\frac{1}{2}\)