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. …

Let us switch to physics for a while with this link to the complete physics courses given by Feynman between 61 and 64 at Caltech. It has historical value, with original audio recordings and photos of the period, but above all pedagogical value, with all a…

In this exercise on Euclidean norms, we needed to know that the set of matrices with determinant \(>0\) is connected. Here's a reminder of the proof: Lemma: Let \(GL_n^+(\mathbb{R})=\{M\in M_n(\mathbb{R}),\det M >0\}\) and \(GL_n^-(\mathbb{R})=\{M\in M_…