# Sum of an Infinite Arithmetic Series

• MHB
• IHateFactorial
In summary, the sum of the infinite arithmetic series \sum_{n=1}^{\infty}n is often mistakenly thought to be equal to -1/12, but this is due to a technique called analytic continuation in the complex plane. The Riemann Zeta function, \zeta (-1), does not converge, but by using analytic continuation, it can be calculated to be equal to -1/12. However, this technique cannot be applied to all expressions and should not be used to determine the sum of infinite series.
IHateFactorial
Somewhere I saw that the sum of the infinite arithmetic series

$$\displaystyle \sum_{n=1}^{\infty}n = \frac{-1}{12}$$

Why exactly is this? I thought infinite arithmetic series had no solution? Also... WHY is it negative? Seems counter-intuitive that the sum of all the NATURAL numbers is a decimal, a negative decimal.

IHateFactorial said:
Somewhere I saw that the sum of the infinite arithmetic series

$$\displaystyle \sum_{n=1}^{\infty}n = \frac{-1}{12}$$

Why exactly is this? I thought infinite arithmetic series had no solution? Also... WHY is it negative? Seems counter-intuitive that the sum of all the NATURAL numbers is a decimal, a negative decimal.
This is a popular topic and there is a lot of misinformation out there about it. As stated the LHS does not exist and certainly wouldn't add up to a fraction on the RHS, much less as a negative value.

This goes a bit deep and can be very confusing. The sum $$\displaystyle \sum_{n = 1}^{\infty} n$$ is a representation of something called the Riemann Zeta function, $$\displaystyle \zeta (-1)$$. The confusion arises because this form of the zeta function cited here does not converge. BUT we can use a technique called "analytic continuation" in the complex plane and write the zeta function in a form where $$\displaystyle \zeta (-1)$$ can be calculated and comes out to -1/12. (The form of the zeta function given by analytic continuation in this domain is not $$\displaystyle \sum_{n = 1}^{\infty} n$$ so we aren't taking an infinite sum.)

This might seem a bit like magic and I need to stress that you cannot do this for just any expression. Analytic continuation does not always increase the size of the domain that you can calculate a function over...the zeta function just happens to be one of them that you can do this with.

-Dan

## 1. What is the formula for finding the sum of an infinite arithmetic series?

The formula for finding the sum of an infinite arithmetic series is S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio.

## 2. How do you know if an infinite arithmetic series has a finite sum?

An infinite arithmetic series will have a finite sum if the common ratio (r) is between -1 and 1. If r is outside of this range, the series will not have a finite sum.

## 3. Can the sum of an infinite arithmetic series be negative?

Yes, the sum of an infinite arithmetic series can be negative if the common ratio (r) is between -1 and 0. This means that the series is decreasing and the sum will approach a negative value.

## 4. What is the difference between a finite and infinite arithmetic series?

A finite arithmetic series has a specific number of terms, while an infinite arithmetic series has an infinite number of terms. The sum of a finite series will have a specific value, while the sum of an infinite series will approach a certain value as the number of terms increases.

## 5. How can the sum of an infinite arithmetic series be used in real life?

The sum of an infinite arithmetic series can be used in many real-life applications, such as calculating the total cost of a loan with compound interest, finding the total distance traveled in a continuously increasing or decreasing speed, and determining the value of an annuity. It is also used in mathematical and scientific fields to model and study continuous processes.

• General Math
Replies
20
Views
2K
• General Math
Replies
7
Views
1K
• General Math
Replies
4
Views
848
• General Math
Replies
3
Views
1K
• General Math
Replies
15
Views
2K
• General Math
Replies
20
Views
1K
• General Math
Replies
1
Views
1K
• Math POTW for University Students
Replies
16
Views
1K
• General Math
Replies
3
Views
1K
• General Math
Replies
7
Views
1K