nLab series

Idea

In analysis a series is a formal precursor to a various notions of a sum of an infinite sequence of numbers.

Definition

An (ordinary) series $\sum_{n=0}^\infty a_n$ whose members are the elements $a_n$ in a given additive group or semigroup is an ordered pair of a sequence $(a_n)_{n=0}^\infty$ and a sequence $(b_k)_k$ of its partial sums $b_k = \sum_{n=0}^k a_k$.

As an operator

For any group or semigroup $G$, we can inductively define the finite sum $\mathbb{N}$-action

$$\sum_{n=0}^{(-)} (-)(n) : (\mathbb{N} \to G) \times \mathbb{N} \to G$$

as

$$\sum_{n=0}^{0} a(n) = a(0)$$

$$\sum_{n=0}^{k+1} a(n) = a(k+1) + \sum_{n=0}^{k} a(n)$$

for all $k:\mathbb{N}$ and $a:\mathbb{N} \to G$, such that currying the action results in the infinite series operator

$$\sum_{n=0}^\infty (-)(n) : (\mathbb{N} \to G) \to (\mathbb{N} \to G)$$

over the function $G$-module $\mathbb{N} \to G$.

Internalisation

In a cartesian closed category $C$, let $(G,+)$ be an additive magma object, let $(N,0,s)$ be a natural numbers object in $C$, and let $a:N\to G$ be an infinite sequence object of elements in $G$. Then there exists an infinite sequence object $b:N\to G$ called the left partial sum infinite sequence object of $a$ inductively defined by $b(0) = a(0)$ and $b(s(n)) = b(n) + a(s(n))$, and an infinite sequence object $c:N\to G$ called the right partial sum infinite sequence object of $a$ inductively defined by $c(0) = a(0)$ and $c(s(n)) = a(s(n)) + c(n)$. The element $b(n)$ is called the left $n$-th partial sum of the infinite sequence $a$, and the element $c(n)$ is called the right $n$-th partial sum of the infinite sequence $a$. A left series object is an infinite sequence object with its left partial sum infinite sequence object, and a right series object is an infinite sequence object with its right partial sum infinite sequence object. If the magma object is commutative and associative, then the left and right partial sum infinite sequence objects of $a$ are equal and just called a partial sum infinite sequence object $d = \sum_{n:N} a(n)$ of $a$, where the element $d(n)$ is the $n$-th partial sum. Then s series object is an infinite sequence object with its partial sum infinite sequence object,

Sum of a series

The most straightforward notion of the sum of a series is the limit of its sequence of partial sums, if this sequence converges, relative to some topology on the space where the members of the sequence belong to. A series that does not converge in this sense is called divergent; sometimes these can also be “summed” by fancier techniques.

Examples

• geometric series

• Mercator series

• Taylor series

• The formal e, $e = \sum_{n=0}^\infty \frac{1}{n!}$, where $\frac{1}{(-)!}:\mathbb{N}\to \mathbb{Q}$

Related concepts

• power series

• asymptotic series

• infinite product

• divergent series

• partial sum

• Riemann series theorem