In this Centrale Supelec problem , we studied the Perron-Frobenius theorem. This theorem has many applications, one of which is at the heart of the Google search engine. We say that \(M>0\) if \(\forall i,j,\quad m_{ij}>0\) et \(\rho(A)=\max\{\abs{\lambda}…

The following exercise will help you apply and better understand Abel transform. It's also an example of a series of functions that converges uniformly on a subset of \(\mathbb{C}\), but not necessarily normally. Exercise: Let \((a_n)_{n\in\mathbb{N…

Comparison theorems are among the fundamental theorems for the study of numerical sequences/series: Theorem: Let \(\sum_{n\in\mathbb{N}} u_n\) and \(\sum_{n\in\mathbb{N}} v_n\) be two series with not negative real terms (the theorem applies more ge…

The Abel transform is an essential technique for studying numerical series: \[ \begin{alignat*}{1} \sum_{k=0}^n u_k &= \sum_{k=0}^n 1\times u_k \\ &= \sum_{k=0}^n (k+1-k)u_k \\ &= \sum_{k=0}^n (k+1)u_k -\sum_{k=1}^n ku_k \\ &= \sum_{k=1}^{n+1} k…

The definition of compact subsets of a metric space is as follows: Définition: Soit \(E\) be a metric space. A subset \(A\subset E\) is compact from any open covering of \(A\) we can find a finite subcovering , i.e. …