Đặt \(A=\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{2014}}\)
\(\Rightarrow3A=1+\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{2013}}\\ \Rightarrow2A=1-\dfrac{1}{3^{2014}}\\ \Rightarrow A=\dfrac{1}{2}-\dfrac{1}{2.3^{2014}}< \dfrac{1}{2}\)
\(M=\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+...+\frac{1}{5^{2013}}+\frac{1}{5^{2014}}\)
Chứng minh rằng \(M< \frac{1}{3}\)
Chứng minh rằng: \(B=\frac{1}{2^3}+\frac{1}{3^3}+...+\frac{1}{2019^3}< \frac{1}{2^2}\)
bài 1: tính A:=\(\frac{1}{2}-\frac{2}{3}+\frac{3}{4}-\frac{4}{5}+\frac{5}{6}-\frac{6}{7}-\frac{5}{6}+\frac{4}{5}-\frac{3}{4}+\frac{2}{3}-\frac{2}{3}-\frac{1}{2}\)
Bài 2: Cho B=\(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+.....+\frac{1}{49}-\frac{1}{50}\)
Chứng minh rằng: \(\frac{7}{12}< A< \frac{5}{6}\)
Chứng minh rằng: \(\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+......................+\frac{99}{100!}< 1\)
Chứng minh rằng: \(\frac{1}{3}+\frac{2}{3^2}+\frac{3}{3^3}+\frac{4}{3^4}+....................+\frac{100}{3^{100}}< \frac{3}{4}\)
Chứng minh rằng : \(\frac{3}{1^2.2^2}+\frac{5}{2^2.3^2}+\frac{7}{3^2.4^2}+...+\frac{19}{9^2.10^2}< 1\)
Chứng minh rằng
\(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{199}-\frac{1}{200}=\frac{1}{101}+\frac{1}{102}+...+\frac{1}{200}\)
bài 1:C=\(\frac{1}{3}+\frac{2}{3^2}+.......+\frac{100}{3^{100}}\)
chứng minh rằng C<\(\frac{3}{4}\)
bài 2
1/ Chứng minh: \(C=\frac{1}{3}-\frac{2}{3^2}+\frac{3}{3^3}-\frac{4}{3^4}....+\frac{99}{3^{99}}-\frac{100}{3^{100}}\). Chứng minh: C < \(\frac{3}{16}\)
