• #1
Svein
Science Advisor
Insights Author
2,176
711
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...


 
  • Like
Likes Greg Bernhardt and Delta2

Answers and Replies

Related Threads on Investigating Some Euler Sums

  • Last Post
Replies
5
Views
4K
Replies
14
Views
2K
  • Last Post
Replies
3
Views
4K
  • Last Post
Replies
5
Views
735
Replies
1
Views
2K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
7
Views
6K
Replies
36
Views
15K
  • Last Post
Replies
5
Views
897
Top