Sum of Natural Numbers (second proof and extra footage)

TL;DR

The sum of all positive integers added together, going to infinity, is equal to -1/12, as proven by the Euler-Riemann zeta function.

Transcript

If I ask you Brady, what's 1 plus 2 plus 3 plus 4 - and if I just keep adding forever - what would you say the answer to that would be?

• (Brady: Well I would say) (it would go- it would tend towards infinity.) Yeah that makes sense, doesn't it? Yeah. (Am I right?) No. It's minus 1/12, it's negative. I've added all these positive numbers together ... Read More

Key Insights

• 🤪 The sum of all positive integers going to infinity is counterintuitively equal to -1/12, as proven using the Euler-Riemann zeta function.
• 🈸 The result has practical applications in physics, particularly in string theory and quantum electrodynamics.
• 🍹 Analytic continuation and careful mathematical manipulation are necessary to understand and obtain meaningful values from divergent sums.
• 🔨 The result challenges our conventional notions of infinity and highlights the importance of mathematical tools in understanding complex concepts.
• 🖐️ The Riemann zeta function, both in its real and complex forms, plays a crucial role in deriving and evaluating the sum of positive integers.
• ♾️ The concept of infinity is not straightforward and requires careful consideration and rigorous mathematical reasoning.
• ❓ The result has connections to the critical dimension in string theory and appears in various packing formulas.

Questions & Answers

Q: How does the sum of all positive integers equal -1/12?

The result seems counterintuitive, but it is proven using the Euler-Riemann zeta function and analytic continuation. By evaluating the function at -1, the sum can be calculated as -1/12.

Q: What practical applications does this result have in physics?

The result is important in various areas of physics, including string theory and quantum electrodynamics. It helps determine the critical dimension in string theory and plays a role in understanding the Casimir force.

Q: Why is it that the sum of positive integers going to infinity is not considered infinite?

Although the sum seems infinite, it is actually a divergent sum that can be regulated and assigned a finite value. Through careful mathematical manipulation, it is possible to obtain a meaningful answer.

Q: Can this result be verified using a calculator by adding up the positive integers?

The result cannot be directly obtained by manually adding up the positive integers on a calculator. The proof involves dealing with infinite sums and divergent numbers, which requires a different approach.

Summary & Key Takeaways

• Despite intuition suggesting that the sum of positive integers going to infinity would be infinite, it is actually equal to -1/12.

• This result has significant implications in physics, particularly in areas such as string theory and quantum electrodynamics.

• The proof of this result involves using the Euler-Riemann zeta function and analytic continuation.