nLab
series

Infinite series

Context

Analysis

Algebra

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

As an operator

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

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

as

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

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

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

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

Internalisation

In a cartesian closed category CC, let (G,+)(G,+) be an additive magma object, let (N,0,s)(N,0,s) be a natural numbers object in CC, and let a:NGa:N\to G be an infinite sequence object of elements in GG. Then there exists an infinite sequence object b:NGb:N\to G called the left partial sum infinite sequence object of aa inductively defined by b(0)=a(0)b(0) = a(0) and b(s(n))=b(n)+a(s(n))b(s(n)) = b(n) + a(s(n)), and an infinite sequence object c:NGc:N\to G called the right partial sum infinite sequence object of aa inductively defined by c(0)=a(0)c(0) = a(0) and c(s(n))=a(s(n))+c(n)c(s(n)) = a(s(n)) + c(n). The element b(n)b(n) is called the left nn-th partial sum of the infinite sequence aa, and the element c(n)c(n) is called the right nn-th partial sum of the infinite sequence aa. 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 aa are equal and just called a partial sum infinite sequence object d= n:Na(n)d = \sum_{n:N} a(n) of aa, where the element d(n)d(n) is the nn-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

Last revised on July 25, 2021 at 03:39:10. See the history of this page for a list of all contributions to it.