a, Cho S=\(\dfrac{1}{\sqrt{1.1998}}+\dfrac{1}{\sqrt{2.1997}}+...+\dfrac{1}{\sqrt{k\left(1998-k+1\right)}}+...+\dfrac{1}{\sqrt{198-1}}\). Hãy so sánh S và 2\(\dfrac{1998}{1999}\)
b, Cho A=\(\dfrac{1}{\sqrt{1.1999}}+\dfrac{1}{\sqrt{2.1998}}+\dfrac{1}{\sqrt{3.1997}}+...+\dfrac{1}{\sqrt{199-1}}\). Hãy so sánh A với 1,999
cho a,b,c > 0 thỏa mãn a + b + c = 6. Chứng minh:
\(\dfrac{a}{\sqrt{b^3+1}}+\dfrac{b}{\sqrt{c^3+1}}+\dfrac{c}{\sqrt{a^3+1}}\ge2\)
Chứng minh \(\dfrac{1}{2\sqrt{1}}+\dfrac{1}{3\sqrt{2}}+\dfrac{1}{4\sqrt{3}}+...+\dfrac{1}{2018\sqrt{2017}}< 2\)
chứng minh rằng nếu \(\dfrac{\sqrt{xy}+1}{\sqrt{y}}=\dfrac{\sqrt{yt}+1}{\sqrt{t}}=\dfrac{\sqrt{xt}+1}{\sqrt{x}}\) thì x=y=t, x.y.t=1
Tìm điều kiện xác định và rút gọn các biểu thức sau :
a/ \(A=\left(\dfrac{\sqrt{3}}{x^2+x\sqrt{3}+3}+\dfrac{3}{x^3-\sqrt{27}}\right).\left(\dfrac{x}{\sqrt{3}}+\dfrac{\sqrt{3}}{x}+1\right)\)
b/ \(B=\dfrac{x^2-\sqrt{x}}{x+\sqrt{x}+1}-\dfrac{x^2+\sqrt{x}}{x-\sqrt{x}+1}+x+1\)
c/ \(C=\left(\dfrac{2+\sqrt{x}}{x+2\sqrt{x}+1}-\dfrac{\sqrt{x}-2}{x-1}\right).\dfrac{x\sqrt{x}+x-\sqrt{x}-1}{\sqrt{x}}\)
d/ \(\left[\dfrac{1}{x-1}+\dfrac{x^2+1-2x}{\left(x-1\right)^2+3x}-\dfrac{1+4x-2x^2}{x^3-1}\right]:\dfrac{2}{x^2+1}\)
cho x,y,z dương thỏa mãn \(\dfrac{1}{xy}+\dfrac{1}{xz}+\dfrac{1}{yz}=1\) tìm max của \(Q=\dfrac{x}{\sqrt{yz\left(1+x^2\right)}}+\dfrac{y}{\sqrt{xz\left(1+y^2\right)}}+\dfrac{z}{\sqrt{xy\left(1+z^2\right)}}\)
Tìm max: \(\dfrac{1}{\sqrt{x}-1}-\dfrac{1}{\sqrt{x}+1}+\dfrac{2x^2+4}{1-x^3}\)
Tìm max:
\(\dfrac{1}{\sqrt{x}-1}-\dfrac{1}{\sqrt{x}+1}+\dfrac{2x^2+4}{1-x^3}\)
a,\(\dfrac{1}{\sqrt{7-\sqrt{24}+1}}\)-\(\dfrac{1}{\sqrt{7+\sqrt{24}-1}}\)
b,\(\dfrac{1}{3-\sqrt{7}}\)-\(\dfrac{1}{3+\sqrt{7}}\)
c,\(\sqrt{21+6\sqrt{6}}\)+\(\sqrt{21-6\sqrt{6}}\)
