Investigating Some Euler Sums

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SUMMARY

The discussion focuses on the investigation of Euler sums, specifically the challenges associated with odd powers compared to even powers, which were resolved by Leonard Euler in the 18th century. The tools available today, such as advanced computational methods, have allowed for precise calculations of odd power sums, yet no general closed-form expression exists for these sums. The harmonic series, represented as 1 + 1/2 + 1/3 + ..., is highlighted as a divergent series, emphasizing the complexity of summing odd powers.

PREREQUISITES
  • Understanding of Euler's work on series and sums
  • Familiarity with harmonic series and its properties
  • Knowledge of Fourier Series applications in mathematical analysis
  • Basic computational methods for numerical analysis
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  • Explore advanced computational techniques for evaluating series
  • Study the properties and applications of harmonic series in mathematics
  • Learn about the historical context of Euler's contributions to mathematics
  • Investigate current research on odd power sums and their implications
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Mathematicians, researchers in number theory, and students interested in advanced series analysis and the historical development of mathematical concepts.

Svein
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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...

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