Transformation d'Abel.
The Abel transform is an essential technique for studying numerical series: \[ \begin{alignat*}{1} \sum_{k=0}^n u_k &= \sum_{k=0}^n 1\times u_k \\ &= \sum_{k=0}^n (k+1-k)u_k \\ &= \sum_{k=0}^n (k+1)u_k -\sum_{k=1}^n ku_k \\ &= \sum_{k=1}^{n+1} ku_{k-1} -\sum_{k=1}^n ku_k \\ &= \sum_{k=1}^{n} k(u_{k-1}-u_k) +(n+1)u_n \\ \end{alignat*} \]
This transformation is also known as integration by discrete parts; indeed, if \[ \begin{alignat*}{1} f:[0,1] &\to \mathbb{R} \\ \end{alignat*} \] is a class \(C^1\) function, and if we let \[ \begin{alignat*}{1} \forall k\in \lBrack 0, n\rBrack,\quad u_k & = f(\frac{k}{n}) \end{alignat*} \] then approximate an integral by a Riemann sum \[ \begin{alignat*}{1} \int_0^1 g(t)\mathrm{d}t &\approx \sum_{k=1}^n\frac{1}{n}g(\frac{k}{n}) \end{alignat*} \] and derivative by \[ \begin{alignat*}{1} f'(\frac{k}{n}) &\approx \frac{f(\frac{k}{n})-f(\frac{k-1}{n})}{\frac{1}{n}} \end{alignat*} \] Then integration by parts \[ \begin{alignat*}{1} \int_0^1 f(t)\mathrm{d}t & = [tf(t)]_0^1 - \int_0^1 tf'(t)\mathrm{d}t \end{alignat*} \] would discretize as follows: \[ \begin{alignat*}{1} \frac{1}{n}\sum_{k=1}^n f(\frac{k}{n}) &\approx \frac{n+1}{n}f(1)-\frac{1}{n}f(0) - \sum_{k=1}^n \frac{k}{n}\frac{f(\frac{k}{n})-f(\frac{k-1}{n})}{\frac{1}{n}}\frac{1}{n} \\ \Leftrightarrow \frac{1}{n}\sum_{k=0}^n u_k &\approx \frac{n+1}{n}u_n - \sum_{k=1}^n \frac{k}{n}(u_k-u_{k-1}) \end{alignat*} \] (in the integration bracket we approach 0 and 1 by respectively \(\frac{1}{n}\) and \(\frac{n+1}{n}\) to recover exactly Abel's transform.)