[[!redirects series]] [[!redirects series operators]] #Contents# * table of contents {:toc} ## Definition ## Given a [[Z-module|$\mathbb{Z}$-module]] $M$ and a sequence $x:\mathbb{N} \to M$ of terms in $M$, the **series operator** $$\Sigma:(\mathbb{N} \to M) \to (\mathbb{N} \to M)$$ is inductively defined as $$\Sigma(x)(0) \coloneqq x(0)$$ $$\Sigma(x)(i + 1) \coloneqq \Sigma(x)(i) + x(i + 1)$$ for $i:\mathbb{N}$. $\Sigma(x)$ is called a **series**. ## See also ## * [[sequence]] * [[inverse series operator]] category: not redirected to nlab yet