Sum of an Infinite Arithmetic Series

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.
  • #1
IHateFactorial
17
0
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.
 
Mathematics news on Phys.org
  • #2
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
 

Related to Sum of an Infinite Arithmetic Series

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.

Similar threads

Replies
20
Views
2K
  • General Math
Replies
7
Views
1K
Replies
4
Views
848
Replies
3
Views
1K
Replies
15
Views
2K
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
Back
Top