Understanding the summation of diverging series

In summary, the conversation discussed the controversial idea that the sum of all natural numbers is -1/12, which is not valid. The concept of assigning a number to a divergent series through summation methods was also brought up, including Abel summation, Cesaro summation, and Ramanujan summation. However, even though ##1+2+3+4+...## can be Ramanujan summed to -1/12, the actual sum of all natural numbers is still infinity. The thread was then closed.
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whit3r0se-
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I was recently researching into some string theory when i came across the following summation:
The sum of all natural numbers is -1/12, now I'm still wrapping my head around the context of the application within critical string dimensions, but is this summation valid? And if not, why it being used in this way?
 
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No, it's not valid. The sum of all natural numbers is infinity, and not -1/12. There are certain ways to assign a number to divergent series however, those ways are called summation methods. You have Abel summation, Cesaro summation, Ramanujan summation, etc. These numbers are not the sums of the series in question. So while ##1+2+3+4+...## can be Ramanujan summed to -1/12, the sum of all natural numbers is still infinity, as it should be.
 
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FAQ: Understanding the summation of diverging series

What is the summation of a diverging series?

The summation of a diverging series is the value obtained by adding all the terms in an infinite series. This value can be either a finite number or infinity, depending on the behavior of the terms in the series.

How can one determine if a series is diverging?

A series is considered to be diverging if the value of its terms increases or decreases without bound as the number of terms increases. This can be determined by examining the behavior of the terms in the series, such as using the ratio test or the comparison test.

Why is it important to understand the summation of diverging series?

Understanding the summation of diverging series is important in various fields of science, such as physics, engineering, and mathematics. It allows us to make accurate predictions and calculations, and helps us understand the behavior of certain phenomena in the natural world.

Can a diverging series have a finite sum?

Yes, it is possible for a diverging series to have a finite sum. This occurs when the terms in the series eventually decrease or increase at a rate slow enough that the series converges to a finite value. However, most diverging series have an infinite sum.

How can one evaluate the summation of a diverging series?

Evaluating the summation of a diverging series can be challenging and may require advanced mathematical techniques. One common method is to use partial sums, where a finite number of terms in the series are added to approximate the sum. Other methods include using advanced calculus techniques, such as integrals and limits.

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