Tính:
\(\left(\dfrac{1}{2}-1\right):\left(\dfrac{1}{3}-1\right):\left(\dfrac{1}{4}-1\right):\) ... : \(\left(\dfrac{1}{50}-1\right)\)
Chứng minh rằng:
\(\left(1+\dfrac{1}{3}+\dfrac{1}{5}+...+\dfrac{1}{50}\right)-\left(\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{6}+...+\dfrac{1}{100}+\dfrac{1}{102}\right)=\dfrac{1}{52}+\dfrac{1}{53}+...+\dfrac{1}{100}+\dfrac{1}{101}+\dfrac{1}{102}\)
CM:\(3< 1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{63}< 6\)
chứng tỏ B = \(1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{63}< 6\)
Cho đẳng thức :\(x\times\left(x+1\right)\times\left(x+2\right)\times.............\times\left(x+2016\right)=2016\)(với x>0)
Chứng tỏ rằng \(x< \dfrac{1}{2015!}\)
cmr :\(\dfrac{1}{2}\)+\(\dfrac{1}{3}\)+\(\dfrac{1}{4}\)+...+\(\dfrac{1}{63}\) > 2
I. tính
1. \(\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{20}+\dfrac{1}{30}\)
2.\(\dfrac{2}{15}+\dfrac{2}{35}+\dfrac{2}{63}+\dfrac{2}{99}+\dfrac{2}{143}\)
II. tìm x
\(1\dfrac{3}{5}+\left(\dfrac{\dfrac{2}{171}}{\dfrac{5}{171}}+\dfrac{\dfrac{2}{373}}{\dfrac{5}{373}}\right)X=\dfrac{16}{5}\)
Chứng minh rằng:
a, A= 1+\(\dfrac{1}{2}\)+ \(\dfrac{1}{3}\)+\(\dfrac{1}{4}\)+......+\(\dfrac{1}{63}\)<6
b, B= \(\dfrac{1}{2}\).\(\dfrac{3}{4}\).\(\dfrac{5}{6}\)......\(\dfrac{9999}{10000}\)< \(\dfrac{1}{100}\)
Tính :
a) \(\dfrac{17}{23}.\dfrac{8}{16}.\dfrac{23}{17}.\left(-80\right).\dfrac{3}{4}\)
b) \(\dfrac{5}{11}.\dfrac{18}{29}-\dfrac{5}{11}.\dfrac{8}{29}+\dfrac{5}{11}.\dfrac{19}{29}\)
c) \(\left(\dfrac{13}{23}+\dfrac{1313}{2323}-\dfrac{131313}{232323}\right).\left(\dfrac{1}{3}+\dfrac{1}{4}-\dfrac{7}{12}\right)\)
d) \(\dfrac{1^2}{1.2}.\dfrac{2^{2^{ }}}{2.3}.\dfrac{3^2}{3.4}.\dfrac{4^2}{4.5}.\dfrac{5^2}{5.6}.\dfrac{6^2}{6.7}.\dfrac{7^2}{7.8}.\dfrac{8^2}{8.9}.\dfrac{9^2}{9.10}\)
e) \(\dfrac{2^2}{3}.\dfrac{3^2}{8}.\dfrac{4^2}{15}.\dfrac{5^2}{24}.\dfrac{6^2}{35}\dfrac{7^2}{48}.\dfrac{8^2}{63}.\dfrac{9^2}{80}\)
BT1: CMR:
a) \(\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{n^2}< 1\)
b) \(\dfrac{1}{4}+\dfrac{1}{16}+\dfrac{1}{36}+\dfrac{1}{64}+\dfrac{1}{100}+\dfrac{1}{144}+\dfrac{1}{196}< \dfrac{1}{2}\)
c) \(\dfrac{1}{3}+\dfrac{1}{30}+\dfrac{1}{32}+\dfrac{1}{35}+\dfrac{1}{45}+\dfrac{1}{47}+\dfrac{1}{50}< \dfrac{1}{2}\)
d) \(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{8}-\dfrac{1}{16}+\dfrac{1}{32}-\dfrac{1}{64}< \dfrac{1}{3}\)
e) \(\dfrac{1}{3}< \dfrac{2}{3^2}+\dfrac{3}{3^3}-\dfrac{4}{3^4}+...+\dfrac{99}{3^{99}}-\dfrac{100}{3^{100}}< \dfrac{3}{16}\)
f) \(\dfrac{1}{41}+\dfrac{1}{42}+\dfrac{1}{43}+...+\dfrac{1}{79}+\dfrac{1}{80}>\dfrac{7}{12}\)
BT2: Tính tổng
a) A=\(\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^3}+...+\dfrac{1}{3^{100}}\)
b) E=\(1+\dfrac{1}{2}\left(1+2\right)+\dfrac{1}{3}\left(1+2+3\right)+\dfrac{1}{4}\left(1+2+3+4\right)+...+\dfrac{1}{200}\left(1+2+3+...+200\right)\)
BT3: Cho S=\(\dfrac{3}{10}+\dfrac{3}{11}+\dfrac{3}{12}+\dfrac{3}{13}+\dfrac{3}{14}\)
CMR: 1 < S < 2
