Ta có: \(M=\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+....+\frac{1}{\sqrt{224}+\sqrt{225}}\)

\(=\sqrt{2}-1+\sqrt{3}-\sqrt{2}+...+\sqrt{225}-\sqrt{224}\)

\(=-1+\sqrt{225}=-1+15=14\)

Và \(N=\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{63}}\)

\(=14,47706...>14=M\)

Giải:

Ta có tính chất tổng quát:

\(\frac{1}{\left(k+1\right)\sqrt{k}+k\left(\sqrt{k+1}\right)}=\frac{\left(k+1\right)\sqrt{k}-k\left(\sqrt{k+1}\right)}{\left(k+1\right)^2k-k^2\left(k+1\right)}\)

\(=\frac{\left(k+1\right)\sqrt{k}-k\left(\sqrt{k+1}\right)}{\left(k+1\right)k\left(k+1-k\right)}=\frac{1}{\sqrt{k}}-\frac{1}{\sqrt{k+1}}\)

Áp dụng vào biểu thức

\(\Rightarrow A=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{224}}-\frac{1}{\sqrt{225}}\)

\(=1-\frac{1}{\sqrt{225}}\)

Với n > 0 ta có:

\(\dfrac{1}{\left(n+1\right)\sqrt{n}+n\sqrt{n+1}}=\dfrac{1}{\sqrt{n}.\sqrt{n+1}.\left(\sqrt{n+1}+\sqrt{n}\right)}=\dfrac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n}.\sqrt{n+1}}=\dfrac{1}{\sqrt{n}}-\dfrac{1}{\sqrt{n+1}}\).

Do đó: \(\dfrac{1}{2+2\sqrt{2}}+\dfrac{1}{\sqrt{3}}-\dfrac{1}{\sqrt{4}}+\dfrac{1}{\sqrt{4}}-\dfrac{1}{\sqrt{5}}+...+\dfrac{1}{\sqrt{224}}-\dfrac{1}{\sqrt{225}}=\dfrac{\sqrt{2}-1}{2}+\dfrac{1}{\sqrt{3}}-\dfrac{1}{\sqrt{225}}=\dfrac{\sqrt{2}-1}{2}+\dfrac{\sqrt{3}}{3}-\dfrac{1}{15}=\dfrac{3\sqrt{2}+2\sqrt{3}-3}{6}-\dfrac{1}{15}=\dfrac{15\sqrt{2}+10\sqrt{3}-17}{30}\)