Infinite Product of a Number

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International Journal of Pure and Applied Mathematical Sciences. ISSN 0972-9828 Volume 10, Number 2 (2017), pp. 107-110 © Research India Publications http://www.ripublication.com

Rushal Sohal

Student The Lexicon International School, Wagholi, Pune, India.

Abstract It seems that multiplying a positive number infinite times would give something extremely huge (infinity), but is it so? In this paper, we will show that any positive integer (>1) raised to the power infinity (i.e. infinite product of n ∞ ∞ =∏푖=1 푛=푛×푛×푛...= 푛 , 푛 > 1) is not infinity; we will use simple mathematical equations and also use some known sums to prove that the infinite product of a positive integer (>1) is not equal to infinity, i.e.∞ 푛∞ ≠ . Keywords: positive integer(퐼+); infinity*(∞); Riemann zeta function 휁(푠).

1. INTRODUCTION So, what is the infinite product of a positive integer (>1), i.e. what is 푛∞? Is it ∞ or something else? Till now, not much research has been done on this topic, and by convention we assume 푛∞ to be equal to ∞. In this paper, we will make simple equations and also use some known sums and prove that 푛∞ = 0 and that, it is not equal to ∞. We are going to approach the problem by observing patterns, assigning and re- assigning variables, and finally solving the equations. But first, we need to prove that 푛∞ ≠ ∞. 2. METHODS The methods for solving the problem are simple but a bit counter-intuitive. In this paper, we will make simple mathematical equations and firstly prove that 푛∞ ≠ ∞. Further in the paper, we will prove that 푛∞ = 0. 2.1 Proving 푛∞ ≠ ∞ Let 푛∞ = ∞, 푛 ∈ 퐼+ − {1} (1) 108 Rushal Sohal

Now, n can be written as sum of 1’s n times, i.e. 푛=1+1+1+. . . (푛 푡푖푚푒푠) Similarly, 푛2 can be written as sum of 1’s 푛2 times, and thus, 푛∞ can be written as: 푛∞ = 1 + 1 + 1+. . . (푛∞ 푡푖푚푒푠) Since, 푛∞ = ∞(푓푟표푚(1)) ⟹ 푛∞ = 1 + 1 + 1+. . . (∞ 푡푖푚푒푠) (2)

2.2 Finding ퟏ + ퟏ + ퟏ +. . . [1] The Riemann zeta function ζ(s) is a function of a complex variable s = σ + it. (The notation s, σ, and t is used traditionally in the study of the zeta function, following Riemann.) ∞ 휁(푠) = ∑ 1⁄푛푠 = 1−푠 + 2−푠 + 3−푠 + . . . 푛=1

[2] Where the sum of n0 occurs in physical applications, it may sometimes be interpreted by zeta function regularization, as the value at s = 0 of the Riemann zeta function ∞ ∞ 휁(푠) = ∑ 1⁄ 푛푠 = (1⁄ 1 − 21−푠) ∑ (−1)푛+1⁄푛푠 푛=1 푛=1 The two formulas given above are not valid at zero however, so one might try the analytic continuation of the Riemann zeta function

휋푠 [3] 휁(푠) = 2푠휋푠−1 sin ( ) Γ(1 − 푠)휁(1 − 푠) 2 Using this one gets (given that Γ (1) = 1),

1 휋푠 1 휋푠 휁(0) = ( ) lim 푠푖푛 ( ) 휁(1 − 푠) = ( ) lim ( − 휋 푠→0 2 휋 푠→0 2 휋3푠3 +. . . ) 48 Where the power series expansion for 휁(푠) about s = 1 follows because 휁(푠)has a simple pole of residue one there. In this sense, 1 1 + 1 + 1+. . . = − (3) 2 [4] The value of 1 + 1 + 1 +. . . can also be found using Bernoulli’s number. For non-positive integers, one has Infinite Product of a Number 109

(−1)푛퐵 휁(−푛) = 푛+1 (B: Bernoulli number) 푛+1 1 1 ⟹ 휁(0) = 퐵 ⁄1 = − (Using the NIST convention, 퐵 = − ) 1 2 1 2

2.3 Substituting the value Therefore, by substituting (3) in (2), we obtain: 1 푛∞ = 1 + 1 + 1+. . . (∞ 푡푖푚푒푠) = − (4) 2 Since, we have assumed 푛∞ = ∞ (푓푟표푚 (1)), 1 ⟹ ∞ = − (푓푟표푚 (1) and (4)) 2 ⟹ 푎 푐표푛푡푟푎푑푖푐푡푖표푛 Thus, 푛∞ ≠ ∞

2.4 Then what is 풏∞? Any positive integer (>1) raised to the power infinity (infinite product) can be written as, 푛∞ = 푛 × 푛 × 푛 ×. . . (∞ 푡푖푚푒푠), 푛 ∈ 퐼+ − {1} (5) Let 푛∞ = 푥 ⟹ 푥 = 푛 × 푛 × 푛 ×. . . (∞ 푡푖푚푒푠) If we observe, the pattern (푛 × 푛 × 푛 ×. . . ) repeats itself after first 푛 and is again equal to 푛∞(= 푥) Thus, we obtain the following equation: 푥 = 푛푥 (6) ⇒ (1 − 푛)푥 = 0 ⇒ 푥 = 0 {( 1 − 푛) ≠ 0 푠푖푛푐푒 푛 ≠ 1(푓푟표푚 (5))}

Thus, 푥 = 푛∞ = 0 One may also think that the solution to equation (6) could be ∞ as infinity multiplied by any positive integer gives infinity, i.e. (∞ × 푛 = ∞). But, the solution ∞ of 푛∞ has already been proven wrong in (2.3).

110 Rushal Sohal

3. RESULTS On solving the above equations, we obtain the result that the infinite product of a positive integer (>1) is zero and not infinity, i.e. 푛∞ = 0 푎푛푑 푛∞ ≠ ∞

4. CONCLUSION What we feel is intuitive or obvious is not always so. It seems that multiplying a number infinitely many times would result ∞, but the fact is, it becomes 0. Till now, there was only an assumption that 푛∞ = ∞, i.e. there was no proof. But, now we do have an equation and a proof and the assumption need not to be assumed. Assumptions/axioms/postulates are assumed true till they aren’t proven wrong or a new theorem is devised.

5. REFERENCES [1] Edward Charles Titchmarsh — The Theory of the Riemann Zeta Function— (Chapter 1-- The function 휁(푠) and the Dirichlet series related to it—Def 1.1) [2] We get the sum of the infinite series- 1 + 1 + 1 +. . ..We reach to this solution via the Riemann zeta function. On solving the Riemann zeta function, we get the sum equal to -1/2. Even evaluating the y-intercept of the graph gives us the answer -1/2. The link for the above is https://en.wikipedia.org/wiki/1_%2B_1_ %2B_1_%2B_1_%2B_%E2%8B%AF [3] Edward Charles Titchmarsh — The Theory of the Riemann Zeta Function— (Chapter 2-- The analytic character of 휁(푠), and the functional equation— Theorem 2.1) [4] Arakawa, Tsuneo, Ibukiyama, Tomoyoshi, Kaneko, Masanobu — Bernoulli Numbers and Zeta Functions

*Infinity (symbol: ∞) is an abstract concept describing something without any bound or larger than any number. In mathematics, "infinity" is often treated as a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as natural or real number.