chứng minh

E

eye_smile

Chứng minh TQ, áp vào bài kia là xong:

~~C/m $\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+...+ \dfrac{1}{\sqrt{n}}<2\sqrt{n}-2$
Có: $\dfrac{1}{\sqrt{k}}<\dfrac{2}{\sqrt{k}+\sqrt{k-1}}=2(\sqrt{k}-\sqrt{k-1})$

\Rightarrow $VT<2(\sqrt{2}-\sqrt{1}+\sqrt{3}-\sqrt{2}+...+\sqrt{n}-\sqrt{n-1})=2\sqrt{n}-2$

~~C/m $\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+...+ \dfrac{1}{\sqrt{n}}>2\sqrt{n}-3$

Có: $\dfrac{1}{\sqrt{k}}>\dfrac{2}{\sqrt{k}+\sqrt{k+1}}=2(\sqrt{k+1}-\sqrt{k})$

\Rightarrow $VT>2(\sqrt{n+1}-\sqrt{n}+...+\sqrt{4}-\sqrt{3}+\sqrt{3}-\sqrt{2})=2\sqrt{n+1}-2\sqrt{2}>2\sqrt{n}-3$
 
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