Series in the nonnegative extended real line

Let $a=(a_k{)}_{k\in \mathbb{N}}$ take values in $[0,+\mathrm{\infty }]\subseteq \bar{\mathbb{R}}$ (The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined). Its partial sums are the unique sequence $(s_n)$ in $[0,+\mathrm{\infty }]$ satisfying

$s_0=0,s_{n+1}=s_n+a_n\mathrm{.}$

To apply The recursion theorem with a fixed successor function, use the state space $\mathbb{N}\times [0,+\mathrm{\infty }]$ and the self-map $T(n,s)=(n+1,s+a_n)$, starting from $(0,0)$. Recursion gives a unique state sequence; induction makes its first coordinate $n$, and its second coordinates are exactly the unique $(s_n)$ satisfying the displayed recurrence. Addition of two nonnegative extended reals is always defined, including when either is $+\mathrm{\infty }$. The sequence $(s_n)$ is nondecreasing, and its nonnegative extended sum is

${\sum }_{k=0}^{\mathrm{\infty }}{a}_k:={\sup }_{n\in \mathbb{N}}{s}_n\in [0,+\mathrm{\infty }],$

whose existence follows from completeness of the extended real line (Every subset of $\bar{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\bar{\mathbb{R}}$, agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$). More generally, for $m\in \mathbb{N}$,

${\sum }_{k\ge m}{a}_k:={\sum }_{j=0}^{\mathrm{\infty }}{a}_{m+j}\mathrm{.}$

Finite sums use the same recursion: ${\sum }_{k<n}{a}_k=s_n$, so the empty sum at $n=0$ is $0$. A double sum such as ${\sum }_i{\sum }_j{a}_{ij}$ means that the inner nonnegative extended sum is formed first and the resulting nonnegative extended sequence is then summed.

The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined

The recursion theorem

Every subset of $\bar{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\bar{\mathbb{R}}$, agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$