Infinite Product

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.

Infinite series is a mathematical expression derived from the mathematical concept of “sequence and series.”. Suppose (a₁, a₂, a₃, a₄,….., a_n) is a sequence of numbers, then the infinite series will be: (a₁a₂a₃…a_n ) or (a₁+a₂+a₃+…+a_n). If we add “+” or “–” to an infinite series, it is called the addition of the infinite series, and if we multiply the numbers, it is called the infinite product or product of the infinite series.

• The addition of infinite series is represented with the sigma notation [Tex]\sum_{i = 0}^{\infty}a_i[/Tex]. Here ” i ” is the index of summation, a_i represents the sequence of the terms, and the relation is taken from i = 0 to i = ∞ .

• The product of infinite series is represented by the pi-product notation [Tex]\prod_{n = 1}^{\infty }[/Tex] [a_n=a₁a₂a₃a₄….] Here a_n represents the sequence of infinite factors, indicating that the series is infinitely continuous.

Infinite series is widely used in disciplines like physics, chemistry, biology, statistics, in engineering, etc. It plays a fundamental role in calculus and other mathematical analysis.

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.

Calculating the product of an infinite series can be more complex than finding its sum, because multiplication tends to be less forgiving when it’s come to convergence. There are several methods to calculate the product of an infinite series:

Infinite Series or Geometric Series Formula

The infinite series formula is S = a/(1-r) . Here a is a number, r is a non zero ratio.

Let us take a = 1/2 and the ratio r = 1/2 , now putting the values,

S = (1/2) / (1-1/2)

S = 1, the solution is 1.

The value of the ratio must lie between -1 to 1 but not 0, i.e., {(-1 < r < 1 ) where r ≠ 0}. If we take r = 6, the series will diverge, and we will not get any definite solution.

Telescoping Series

Some types of infinite series have telescoping property. If you can express the numbers in such a way that many of them can cancel out each other when you multiply them, this property is known as telescoping property of infinite series. By this process you can compute the product of infinite series.

Ratio Test

For some infinite series, you can use ratio test to determine whether the series is convergence or divergence. There is a certain limit of the ratio of consecutive terms to find is it convergence or divergence. If the ratio is less than 1, the series converges and if the ratio is above 1 then the series diverges. It is quite related to infinite series formula. By this process you may be able to compute the product of an infinite series.

If lim_(n→∞)⁡ a_n+1/a_n = r, then,

• If r < 1 , the series absolutely converges
• If r = 1 , the result is inconclusive
• If r > 1 , the series absolutely diverges.

Cauchy Product

The cauchy product is applied to calculate the product of infinite series. It converges absolutely. If two infinite series converge absolutely, then their product is said to be equal to the cauchy product of the two series.

• Cauchy Product of Two Infinite Series:

Let [Tex]\sum_{i = 0}^{\infty } a_i[/Tex] and [Tex]\sum_{n = 0}^{\infty } a_n[/Tex] are two infinite series. The Cauchy product of these two infinite series is defined as:

[Tex]((\sum_{i = 0}^{\infty }a_i)¦) · ((\sum_{i = 0}^{\infty } b_n)¦) = \sum_{k = 0}^{\infty } c_k[/Tex] where c_k = [Tex]\sum_{i = 0}^{k } a_mb_{k-m}[/Tex]

It is to remember that, not all the infinite series have well defined product. After multiplication, some series diverge, some converge. So, having the knowledge of convergence is very important for the students to compute the product of infinite series.

Some of the solved problems on the product of an infinite series are discussed below:

Example 1. Discuss the convergence of the infinite product [Tex]\prod_{n = 1}^{\infty } (1+1/n^2)[/Tex]

Solution:

The given product = [Tex]\prod_{n = 1}^{\infty } (1+1/n^2)[/Tex]

= [Tex] \prod_{n = 1}^{\infty } (1+a_n)[/Tex], where a_n = 1/n² ≥ 0 for all n.

∴ [Tex]\prod_{n = 1}^{\infty } (1+a_n)[/Tex] and [Tex]\sum_{n = 1}^{\infty } a_n[/Tex] converge or diverge together.

But [Tex]\sum_{n = 1}^{\infty } a_n = \sum_{n = 1}^{\infty } 1/n^2[/Tex] is convergent by p-test,

then the infinite product, [Tex]\prod_{n = 1}^{\infty } (1+a_n) = \prod_{n = 1}^{\infty } (1+1/n^2)[/Tex] is also convergent.

Example 2. Discuss the convergence of the infinite product \prod_{n = 2}^{\infty } (1-1/√n).

Solution:

Given product = [Tex]\prod_{n = 2}^{\infty } (1-1/\sqrt n)[/Tex]

= [Tex]\prod_{n = 2}^{\infty } (1-b_n)[/Tex], where b_n = 1/√n , n ≥ 2 , (0 ≤ bn <1)

∴ [Tex]\prod_{n = 2}^{\infty } (1-b_n)[/Tex] and [Tex]\sum_{n = 2}^{\infty } b_n[/Tex] converge and diverge together.

But [Tex]\sum_{n = 2}^{\infty } b_n = \sum_{n = 2}^{\infty } \frac{1}{\sqrt n}[/Tex] is divergent by p-test,

then the infinite product, [Tex]\prod_{n = 2}^{\infty } (1-b_n) = \prod_{n = 2}^{\infty } (1-\frac {1}{\sqrt n})[/Tex] also diverges to zero.

Example 3. Find the value of the infinite series when the first term is 2/3 and the common ratio is 5.

Solution:

Here we will use the formula S=a/(1-r) . But note that the common ratio is 5 which is greater than 1, hence the series diverges and does not possess any solution. The common ratio must lie between -1 and 1. But the ratio should not be zero.

Example 4. Discuss the convergence of the infinite product (1 + 1/√2)(1 – 1/√3)(1 + 1/√4)(1 – 1/√5) ….

Solution:

Given product = (1 + 1/√2)(1 – 1/√3)(1 + 1/√4)(1 – 1/√5) …

= [Tex]\prod_{n = 2}^{\infty } [1+\frac {(-1)^n}{\sqrt n}][/Tex]

= [Tex]\prod_{n = 2}^{\infty } (1+a_n)[/Tex], where a_n = (-1)ⁿ/√n and (a_n )² = 1/n

Now, [Tex]\sum_{n = 2}^{\infty } a_n[/Tex] is convergent but [Tex]\sum_{n = 2}^{\infty } a_n^2[/Tex] is divergent by p-test,

∴ Infinite product [Tex]\prod_{n = 2}^{\infty } [1+\frac {(-1)^n}{\sqrt n}][/Tex] diverges to zero.

Also, Check

• Number Theory
• Sequence and Series
• Geometric Progression

What does П mean?

The symbol П is called pi. It is used to denote the product of infinite series. for example, 4Пi=1.2.3.4=24 .

What is the infinite series formula?

The formula for the infinite series is S=a/(1-r), where a is the first term and r is the common ratio.

What is the infinite product of a sequence?

An infinite product is a sequence of numbers (usually real or complex ( a _k ) _{k ∈ N} written as ∏_(k=0)^∞ a_k .

What is the meaning of ∑ ?

The symbol ∑ is known as sigma. It is used to denote the sum of an infinite series.

What is infinity?

Infinity is a concept, not a number. You can make an approach to reach infinity by counting higher and higher values, but you will never actually reach to it.

Who invented the symbol of infinity?

The symbol of infinity is ∞ , a horizontal 8. It was discovered by the British mathematician John Wallis in 1655.

Can an infinite product be zero?

The value of a Convergent infinite product can be zero, if and only if finite number of factors of the sequence is zero.