Investigating Some Euler Sums

[Svein]

So, why only odd powers? Mostly because the even powers were solved by Leonard Euler in the 18th century. Since the “mathematical toolbox” at that time did not contain the required tools, he needed 6 years to prove the validity of his deductions. Now, however, we have much more powerful tools available, as I have shown in one of my previous insights (Using the Fourier Series to Find Some Interesting Sums).
Leaving the even powers aside, the odd powers are much more difficult. Using a computer, the values have been calculated to an awesome degree of precision, but as far as I know, no general closed-form expression has been found.
The first odd positive number is of course 1. The corresponding series is of course 1+1/2+1/3+⋯. This series has a special name – the harmonic series. Unfortunately, this sum diverges, albeit very slowly. It can be shown that the partial sum up to 1/N tends to ln⁡(N)+γ, where γ is the Euler-Mascheroni constant (approximately 0.5772). Therefore, we will...

Continue reading...

 

[jvicens]

As a fellow scientist, I appreciate your curiosity about the odd powers and the history behind their exploration. You are correct in pointing out that the even powers were thoroughly studied by Euler in the 18th century, and that he faced significant challenges in proving their validity due to the limited mathematical tools available at the time. However, with the advancement of technology and the development of more powerful mathematical tools, we have been able to make great strides in understanding and solving problems related to the odd powers.

The fact that the values for the odd powers have been calculated to a high degree of precision using computers is a testament to the progress we have made in this area. However, as you mentioned, a general closed-form expression for the odd powers has not been found yet. This is because the odd powers are much more complex and difficult to work with compared to the even powers. They pose unique challenges and require innovative approaches to solve them.

The harmonic series, which is the series corresponding to the first odd positive number, 1, is a perfect example of the complexity of the odd powers. While the series does not converge, it has been shown that the partial sum up to 1/N approaches ln(N)+γ, where γ is the Euler-Mascheroni constant. This shows that even though the series does not have a finite sum, it does have a well-defined limit that can be expressed in terms of other important mathematical constants.

In conclusion, while the even powers may have been thoroughly studied and solved in the past, the odd powers continue to intrigue and challenge us as scientists. With the powerful tools and technology at our disposal, I have no doubt that we will continue to make progress in understanding and solving problems related to the odd powers. Thank you for bringing this topic to our attention and for sharing your insights on using the Fourier Series to find interesting sums.