What is up with people thinking that 1 + 2 + 3 + = -1/12?

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In summary, the sum 1 + 2 + 3 + ... is not equal to -1/12 in the traditional sense of addition. However, in certain branches of mathematics, such as number theory and string theory, it can be interpreted using a technique called zeta function regularization, which assigns a finite value to divergent series. This results in the negative value of -1/12 for the sum of an infinite series, rather than a traditional addition of finite numbers. While this concept has been used in various fields of mathematics, it is not directly applicable in the real world and is simply a tool to make certain infinite series meaningful. The Riemann zeta function is often associated with this sum, as it is used to
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Jamin2112
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What is up with people thinking that 1 + 2 + 3 + ... = -1/12?

My friend showed me a Youtube video where a Physicist "proves" that ∑kεNk = -1/12. The guy in the video uses a lot of illegal maneuvers. But apparently this "fact" is used in String Theory. Do Physicists use different definitions of convergence or something? I can easily prove that the sequence {1 + 2 + ... + n} is unbounded using the definitions I learned in Intro to Real Analysis in college.
 
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We had a recent thread on this.

tl;dr The manipulations are indeed invalid. But physicists do use the Zeta regularization in their work.
 

1. What is the significance of the sum 1 + 2 + 3 + ... equaling -1/12?

The sum 1 + 2 + 3 + ... is not equal to -1/12 in the traditional sense of addition. However, in certain branches of mathematics, such as number theory and string theory, this sum can be interpreted using a mathematical concept called zeta function regularization. This technique assigns a finite value to divergent series, such as 1 + 2 + 3 + ..., in order to make them mathematically meaningful.

2. How can 1 + 2 + 3 + ... possibly equal a negative number?

Again, it is important to note that this sum is not being treated as a traditional addition problem. The negative value of -1/12 is a result of using zeta function regularization, which is a mathematical tool used to assign finite values to divergent series. In this context, the negative value represents the sum of an infinite series, rather than a traditional addition of finite numbers.

3. Is this just a mathematical trick or does it have real-world applications?

Zeta function regularization has been used in various fields of mathematics, such as number theory and string theory, to help solve complex problems. However, the concept of 1 + 2 + 3 + ... equaling -1/12 is not directly applicable in the real world. It is simply a mathematical tool used to make certain infinite series meaningful.

4. How is this sum related to the Riemann zeta function?

The Riemann zeta function is a mathematical function that can be used to express the sum of an infinite series, such as 1 + 2 + 3 + ..., using zeta function regularization. When the value of the Riemann zeta function is evaluated at -1, it results in the finite value of -1/12, which is why this sum is often associated with the Riemann zeta function.

5. Can you explain how zeta function regularization works in more detail?

Zeta function regularization involves assigning a finite value to a divergent series by using a mathematical concept called analytic continuation. This technique involves extending the domain of a function beyond its original definition in order to assign values to previously undefined or infinite points. In the case of 1 + 2 + 3 + ..., the sum is expressed as the value of the Riemann zeta function at -1, which is obtained through analytic continuation.

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