Find a finite sum Find the following sum :
$\frac{1}{1^2}+ \frac{1+2}{1^2 +2^2}+\frac{1+2+3}{1^2 +2^2 +3^2}+...+\frac{1+2+3+....n}{1^2 +2^2 +3^2 +....+n^2}$
I tried to unify the denominators but i got a complicated form so i could not go on ?

We obtain using the finite summation formulas
\begin{align*}
  \sum_{k=1}^nk&=\frac{1}{2}n(n+1)\qquad\text{and}\qquad\sum_{k=1}^nk^2=\frac{1}{6}n(n+1)(2n+1)\\
  \end{align*}
and the harmonic numbers $H_n=1+\frac{1}{2}+\cdots+\frac{1}{n}$

\begin{align*}
\color{blue}{\sum_{j=1}^n\frac{1+2+\cdots+j}{1^2+2^2+\cdots+j^2}}
&=\sum_{j=1}^n\frac{\frac{1}{2}j(j+1)}{\frac{1}{6}j(j+1)(2j+1)}\\
&=3\sum_{j=1}^n\frac{1}{2j+1}\\
&=3\left(\sum_{j=1}^{2n+1}\frac{1}{j}-\sum_{j=1}^{n}\frac{1}{2j}\right)\\
&\,\,\color{blue}{=3\left(H_{2n+1}-\frac{1}{2}H_n\right)}
\end{align*}

