I know that I have to use the inverse of the Hilbert matrix, but I can understand how.
Let $x_{1},\dots,x_{N}$ real numbers such that $$\underset{i=1}{\overset{N}{\sum}}\frac{x_{i}}{i+ j}=\frac{1}{2j+1}$$ for each natural number $j\in\left[1,N\right]$. Find (in function of $N$) the value of $$\underset{i=1}{\overset{N}{\sum}}\frac{x_{i}}{2i +1}.$$
To start, let me say that I don't know how to solve this problem.
I arrived at the answer by checking the results for some small values of n and then making a wild guess.
If you have an explicit expression for the i-j entry of the inverse of H then that would be one way.
There is such a formula for the inverse of the Hilbert matrix in terms of binomial coefficients and it is quite complicated.
Another way would be to use induction on the size of H.
I'm not able to use the explicit inverse of the Cauchy (Hlibert) matrix. This is a new "hint" for this problem: if you consider the Shifted Legendre polynomials $$P_{n}\left(x\right)=\frac{1}{n!}\frac{d^{n}}{dx^ {n}}\left(x-x^{2}\right)^{n}$$ you have $$\int_{0}^{1} P_{n}\left(x\right)P_{m}\left(x\right)dx=\frac{ \delta_{mn} }{2n+1}$$ where $\delta_{mn}$ is the Kronecker delta. How can use it? Can someone help me?