Tính tỉ số \(\frac{A}{B}\), biết:
A = \(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2007}+\frac{1}{2008}+\frac{1}{2009}\)
B = \(\frac{2008}{1}+\frac{2007}{2}+\frac{2006}{3}+...+\frac{2}{2007}+\frac{1}{2008}\)
a,A=1-2+3+4-5-6+7+8-9-...+2007+2008-2009-2010
b, \(\frac{1}{5^2}-\frac{1}{5^3}+\frac{1}{5^4}-\frac{1}{5^5}+..-\frac{1}{5^{101}}\).CM<\(\frac{1}{30}\)
Bài 1 : Cho N =\(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2009^2}+\frac{1}{2010^2}\)
Hãy chứng minh rằng N<1
\(Cho\) \(A=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2018^2}\)
\(Chứng\) \(minh\) \(\frac{2017}{2018}>A>\frac{2008}{2018}\)
Tìm số tự nhiên x, biết:
\(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{2}{x.\left(x+1\right)}=\frac{2009}{2010}\)
1.so sánh
a. A=\(\frac{2005^{2005}+1}{2005^{2006}+1}\) và B=\(\frac{2005^{2004}+1}{2005^{2005}+1}\)
b. A=\(\frac{20^{10}+1}{20^{10}-1}\) và B=\(\frac{20^{10}-1}{20^{10}-3}\)
c. A=\(\frac{2009^{2009}+1}{2009^{2010}+1}\) và B=\(\frac{2009^{2010}+2}{2009^{2011}-2}\)
d. A=\(\frac{2013^{2014}+2014}{2013^{2014}-2014}\) và B=\(\frac{2013^{2014}-2014}{2013^{2014}-6042}\)
\(\frac{10+\frac{9}{2}+\frac{8}{3}+\frac{7}{4}+ \frac{6}{5}+\frac{5}{6}+\frac{4}{7}+\frac{3}{8}+\frac{2}{9}+\frac{1}{10}}{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{10}+\frac{1}{11}}\)
Tính giá trị của :
D=\(\left(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{2019^2}\right)x\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2020^2}\right)-\left(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{2020^2}\right)x\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2019^2}\right)\)
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) \(\frac{1}{3}-\frac{2}{3^2}+\frac{3}{3^3}-\frac{4}{3^4}+.....+\frac{99}{3^{99}}-\frac{100}{3^{100}}< \frac{3}{16}\)
