$2.\;$ Ta có các tính chất sau (chứng minh bằng quy nạp)$1+3+5+\ldots+(2n+1)=n^2$
$1+2+3+\ldots+n=\frac{n(n+1)}{2}\Rightarrow \frac 1{1+2+3+\ldots+n}=\frac 2n-\frac 2{n+1}$
$P=\frac{1}{\sqrt{1^2}}+\frac{1}{\sqrt{1^2}+\sqrt{2^2}}+\frac{1}{\sqrt{1^2}+\sqrt{2^2}+\sqrt{3^2}}+\ldots\frac{1}{\sqrt{1^2}+\sqrt{2^2}+\ldots+\sqrt{n^2}}$
$\quad=\frac 11+\frac 1{1+2}+\frac 1{1+2+3}+\ldots+\frac 1{1+2+3+\ldots+n}$
$\quad=\frac 21-\frac 22+\frac 22-\frac 23+\frac 23-\frac 24+\ldots+\frac 2{n}-\frac 2{n+1}$
$\quad=\boxed{\frac{2n}{n+1}}$