Câu 1:
\(\dfrac{1}{a}-\dfrac{1}{a+1}< \dfrac{1}{a^2}\)(1)
=>\(\dfrac{a+1-a}{a\left(a+1\right)}< \dfrac{1}{a^2}\)
=>\(\dfrac{1}{a^2+a}< \dfrac{1}{a^2}\)
=>a^2+a>a^2
=>a>0
mà a>1
nên (1) đúng
\(\dfrac{1}{a^2}< \dfrac{1}{a-1}-\dfrac{1}{a}\)(2)
=>\(\dfrac{1}{a^2}< \dfrac{a-a+1}{a\left(a-1\right)}=\dfrac{1}{a\left(a-1\right)}\)
=>a^2>a^2-a
=>a^2-a<a^2
=>-a<0
=>a>0
mà a>1
nên (2) đúng
=>\(\dfrac{1}{a}-\dfrac{1}{a+1}< \dfrac{1}{a^2}< \dfrac{1}{a-1}-\dfrac{1}{a}\)
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