# Astounding: 1+2+3+4+5+ = [not infinity]

• FlexGunship
In summary, there is a surprising and counter-intuitive result that the sum of an infinite series can equal a specific number, even if the series does not have a limit. This idea is uncomfortable for many people, but it has practical applications in areas such as the Casimir force and string theory. To understand this concept, rigorous application of science and math is necessary, as demonstrated in various videos. There is a thread on this topic in the General Math forum.
FlexGunship
Gold Member
$\displaystyle\sum_{n=1}^{\infty}{n} = {-}\dfrac{1}{12}$

It seems that, regardless of intelligence, this proof (or sum, or proof of sum, or demonstration of sum) rattles some more than others. It is so astoundingly counter-intuitive that embracing it means really leaving your mathematical comfort zone.

Most objections to the resolution of this sum come from how uncomfortable people become when the sum of a series is not directly related to it's limit (or there is no limit, but there exists a sum).

I'll admit I'm not a PhD mathematician, so I'm not sufficiently skilled to dispute this on a mathematical level; however, I accept that algebraic operations can be performed on this infinite sum if for no other reason than that both the Casimir force (experimentally demonstrated) and string theory's critical dimension calculation (yet to be demonstrated) both require it.

This is the first time in my life that I've considered getting a tattoo. What a great reminder that reality is way more interesting than our intuition can handle. And FURTHER, that the only way for us to get our primate-brains to understand it is through rigorous application of science and math!

Here are the videos I had to watch before being even marginally convinced:

1. Proof of Grandi's series
2. Algebraic Demonstration
3. Zeta-function Proof

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## 1) What is the answer to "Astounding: 1+2+3+4+5+ = [not infinity]"?

The answer is that the sum of 1+2+3+4+5 is actually a finite number, which is 15. This may seem counterintuitive since we are used to thinking of infinity as the sum of an infinite series, but in this case, we are only adding a finite number of terms.

## 2) How can the sum of a finite number of terms be a finite number, instead of infinity?

This is because the sum of a finite series can be calculated using a formula, such as the formula for the sum of consecutive integers (n(n+1)/2). In this case, the formula for the sum of 1+2+3+4+5 is (5(5+1)/2) = 15. So even though we are adding multiple terms, we are still using a finite process to arrive at a finite answer.

## 3) Why is this concept of "not infinity" in the equation "Astounding: 1+2+3+4+5+ = [not infinity]" important?

This concept is important because it challenges our understanding of infinity and highlights the fact that not all series or sums are infinite. It also shows that mathematical concepts and equations can sometimes have unexpected or counterintuitive results.

## 4) Is this idea of a finite sum applicable to other series or sums?

Yes, this idea can be applied to other series or sums that have a finite number of terms. As long as we can find a formula or method to calculate the sum, it will result in a finite answer. However, there are also series or sums that are truly infinite and do not have a finite answer.

## 5) How does this concept tie into the overall study of mathematics?

This concept ties into the overall study of mathematics by challenging our understanding and assumptions about basic mathematical concepts, such as infinity. It also highlights the importance of carefully defining and understanding mathematical terms and symbols, as they can lead to unexpected results if not properly understood.

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