Chapter 1: Understanding Infinite Sums

When we sum numbers, each term increasing by a fixed positive amount, we naturally expect the total to grow larger. For instance, the sum of all natural numbers from 1 to 10 is 55, and from 1 to 100, it reaches 5050. As we extend this to infinity, one would assume the sum approaches infinity. However, through specific calculations, we arrive at the unexpected result of -1/12. This outcome is perplexing—how can the addition of positive numbers yield a negative fraction? This article aims to unravel the reasoning behind this mathematical anomaly.

To begin, let’s examine some other summation patterns. Consider the series: 1 - 1 + 1 - 1 + 1 - 1… This infinite series alternates between ones and minus ones. If we could determine the final sign of this series, we could easily deduce its sum. For convenience, let’s label this series as A. By isolating the negative sign after the first term, we see that since the series is infinite, the expression in parentheses equals A again. Through straightforward algebra, we find that A = ½, specifically, 1 - 1 + 1 - 1 + 1 - 1… = 1/2.

Next, we’ll analyze the series: 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 +… We’ll call this series B. If we add the previously computed series A = ½ but start at the second term, we can visualize it as follows. This leads to subtracting the same series B from 1. Through regular algebraic manipulation, we conclude that B = ¼.

Finally, we arrive at the essential series— the sum of all natural numbers, which we will designate as X. Thus, X = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 +… If we subtract the series B = ¼ from this, we get the expression 1 - 1 + 2 + 2 + 3 - 3 + 4 + 5 - 5 + 6 + 6 + 7 - 7… For clarity, let's rewrite this. Eventually, we find that our sum of all natural numbers, X, minus ¼ leads us to 4 + 8 + 12 + 16… which is a sum of all multiples of four. Factoring out the 4 gives us 4(1 + 2 + 3 + 4…), which brings us back to series X, leading to the conclusion that X = -1/12. Thus, the sum of all natural numbers is indeed -1/12.

Interestingly, this seemingly paradoxical result has significant implications in physics. It plays a crucial role in quantum mechanics, appearing in various studies related to string theory, and has been instrumental in explaining the Casimir Effect. While the notion of the sum of all natural numbers equating to a negative value raises concerns regarding our summation methods, the repeated application of -1/12 in scientific literature—with consistent and accurate results—underscores its legitimacy.

A rigorous proof demonstrating why the sum of all natural numbers equals -1/12.

Chapter 2: The Mathematical Journey

An exploration of the intriguing argument for -1/8 as the sum of all natural numbers.