Violympic toán 9
Các câu hỏi tương tự
cmr:
\(\dfrac{1}{9}+\dfrac{1}{16}+...+\dfrac{1}{\left(2n+1\right)^2}< \dfrac{1}{4}\left(\forall n\ge1\right)\)
Tính các tổng sau:
Adfrac{1}{2.5}+dfrac{1}{5.8}+dfrac{1}{8.11}+.....+dfrac{1}{left(3n-1right)left(3n+2right)}
Bdfrac{1}{1.3.5}+dfrac{1}{3.5.7}+dfrac{1}{5.7.9}+....+dfrac{1}{left(2n-1right)left(2n+1right)left(2n+3right)}
Csqrt{1+dfrac{1}{1^2}+dfrac{1}{2^2}}+sqrt{1+dfrac{1}{2^2}+dfrac{1}{3^2}}+sqrt{1+dfrac{1}{3^2}+dfrac{1}{4^2}}+....+sqrt{1+dfrac{1}{2018^2}+dfrac{1}{2019^2}}
Đọc tiếp
Tính các tổng sau:
A=\(\dfrac{1}{2.5}+\dfrac{1}{5.8}+\dfrac{1}{8.11}+.....+\dfrac{1}{\left(3n-1\right)\left(3n+2\right)}\)
B=\(\dfrac{1}{1.3.5}+\dfrac{1}{3.5.7}+\dfrac{1}{5.7.9}+....+\dfrac{1}{\left(2n-1\right)\left(2n+1\right)\left(2n+3\right)}\)
C=\(\sqrt{1+\dfrac{1}{1^2}+\dfrac{1}{2^2}}+\sqrt{1+\dfrac{1}{2^2}+\dfrac{1}{3^2}}+\sqrt{1+\dfrac{1}{3^2}+\dfrac{1}{4^2}}+....+\sqrt{1+\dfrac{1}{2018^2}+\dfrac{1}{2019^2}}\)
4 tháng 12 2021 lúc 16:22
Cho \(A_n=\dfrac{1}{\left(2n+1\right)\sqrt{2n-1}},\forall n\in N\text{*}\)
CMR: \(A_1+A_2+...+A_n< 1\)
Tính các tích sau:
P_1 left(1+dfrac{2}{4}right)left(1+dfrac{2}{10}right)left(1+dfrac{2}{18}right)....left(1+dfrac{2}{n^2+3n}right)
P_2 left(1+dfrac{1}{3}right)left(1+dfrac{1}{8}right)left(1+dfrac{1}{15}right)....left(1+dfrac{2}{n^2+2n}right)
P_3 left(1-dfrac{1}{1+2}right)left(1-dfrac{1}{1+2+3}right)left(1-dfrac{1}{1+2+3+4}right).....left(1-dfrac{1}{1+2+3+4+...+n}right)
P_4 dfrac{2^4+4}{4^4+4}.dfrac{6^4+4}{8^4+4}.dfrac{8^4+4}{10^4+4}....dfrac{18^4+4}{20^4+4}
Đọc tiếp
Tính các tích sau:
P\(_1\) =\(\left(1+\dfrac{2}{4}\right)\left(1+\dfrac{2}{10}\right)\left(1+\dfrac{2}{18}\right)....\left(1+\dfrac{2}{n^2+3n}\right)\)
P\(_2\) =\(\left(1+\dfrac{1}{3}\right)\left(1+\dfrac{1}{8}\right)\left(1+\dfrac{1}{15}\right)....\left(1+\dfrac{2}{n^2+2n}\right)\)
P\(_3\) = \(\left(1-\dfrac{1}{1+2}\right)\left(1-\dfrac{1}{1+2+3}\right)\left(1-\dfrac{1}{1+2+3+4}\right).....\left(1-\dfrac{1}{1+2+3+4+...+n}\right)\)
P\(_4\) = \(\dfrac{2^4+4}{4^4+4}.\dfrac{6^4+4}{8^4+4}.\dfrac{8^4+4}{10^4+4}....\dfrac{18^4+4}{20^4+4}\)
CMR, ∀n ≥ 1, n ∈ N : \(\dfrac{1}{2}\)+\(\dfrac{1}{3\sqrt{2}}\)+\(\dfrac{1}{4\sqrt{3}}\)+....+ \(\dfrac{1}{\left(n+1\right)\sqrt{n}}\)<2
Với số tự nhiên n, \(n\ge3\). Đặt \(S_n=\dfrac{1}{3\left(1+\sqrt{2}\right)}+\dfrac{1}{5\left(\sqrt{2}+\sqrt{3}\right)}+...+\dfrac{1}{\left(2n+1\right)\left(\sqrt{n}+\sqrt{n+1}\right)}\). Chứng minh: \(S_n< \dfrac{1}{2}\)
a)Cho 0 < c ; c < b ; b < a . CMR:\(\sqrt{c\left(a-c\right)}+\sqrt{b\left(b-c\right)}\le\sqrt{ab}\)
b)Cho \(x\ge1;y\ge1\). CMR:\(\dfrac{1}{1+x^2}+\dfrac{1}{1+y^2}\ge\dfrac{2}{1+xy}\)
cho dãy số:
\(a_1=1,a_2=1+\dfrac{1}{3},...,a_n=1+\dfrac{1}{3}+\dfrac{1}{5}+...+\dfrac{1}{2n-1}\)
cmr:\(\dfrac{1}{a_1^2}+\dfrac{1}{3a_2^2}+...+\dfrac{1}{\left(2n-1\right)a_n^2}< 2\)
cho dãy số :\(a_1=1,a_2=1+\dfrac{1}{3},.....,a_n=1+\dfrac{1}{3}+\dfrac{1}{5}+...+\dfrac{1}{2n-1}\)
cmr:
\(\dfrac{1}{a_1^2}+\dfrac{1}{3a_2^2}+...+\dfrac{1}{\left(2n-1\right)a_n^2}< 2\)