Product of Infinite Series or Infinite Product

The product of an infinite number of factors given in a sequence, following some definite law is called the infinite product.

If ❮a_n❯ is a sequence, then (a₁a₂a₃a₄…a_n…) is an infinite product. It is denoted by-

[Tex]\prod_{n = 1}^{\infty } a_n[/Tex]

thus,

[Tex]\prod_{n = 1}^{\infty } a_n[/Tex] = a₁a₂a₃a₄a₅….a_n.. is an infinite series.

Partial Products of Infinite Product

If [Tex]\prod_{n = 1}^{\infty } a_n[/Tex] is an infinite product then the sequence ❮P_n❯ , where

P_n = [Tex]\prod_{n = 1}^{\infty } a_n[/Tex] = a₁a₂a₃a₄… , here a_n is called the sequence of partial products of infinite product.

P₁ = a₁

P₂ = a₁a₂

P₃ = a₁a₂a₃

……

P_n = a₁a₂….a_n

here P_n is the nth partial product.

Convergence of Infinite Product

Let P_n = a₁a₂a₃a₄….a_n be the nth partial product of [Tex]\prod_{n = 1}^{\infty } a_n[/Tex]

• if no factor of the above sequence is zero and lim_(n→∞)⁡P_n = P, (P ≠ 0) where P is a finite value. Then,

[Tex]\prod_{n = 1}^{\infty } a_n = P[/Tex]

• if P = ∞ , then the infinite product,

[Tex]\prod_{n = 1}^{\infty } a_n[/Tex] is said to diverge to ∞ .

• if P = 0 or infinitely many factors are zero, then the infinite product,

[Tex]\prod_{n = 1}^{\infty } a_n[/Tex] is said to diverge to 0 .

• If finite factors of the sequence are zero, then the infinite product [Tex]\prod_{n = 1}^{\infty }[/Tex] is said to converge, if it converges after removing all the factors which are zero.

Infinite product is a mathematical expression that represents the product of an infinite sequence of terms. Infinite series is a mathematical expression of an infinite number of terms which are arranged in a specific relation like sum or products. This number of terms are mainly in the form of numbers, functions and quantities. For example, ( a₁+a₂+a₃+a₄+…+a_n+.. ) is an infinite series. It has applications in disciplines such as mathematics, physics, chemistry, biology, computer science, etc.

In this article, we will discuss in detail the infinite series, the product of the infinite series, its calculations, and other topics related to it.