analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
…
…
symmetric monoidal (∞,1)-category of spectra
In analysis a series is a formal precursor to a various notions of a sum of an infinite sequence of numbers.
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 $b_k = \sum_{n=0}^k a_k$.
For any group or semigroup $G$, we can inductively define the finite sum $\mathbb{N}$-action
as
for all $k:\mathbb{N}$ and $a:\mathbb{N} \to G$, such that currying the action results in the infinite series operator
over the function $G$-module $\mathbb{N} \to G$.
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,
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.
The formal e, $e = \sum_{n=0}^\infty \frac{1}{n!}$, where $\frac{1}{(-)!}:\mathbb{N}\to \mathbb{Q}$
Last revised on January 4, 2023 at 17:27:15. See the history of this page for a list of all contributions to it.