Insights Investigating Some Euler Sums

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    Euler Sums
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The discussion focuses on Euler sums, particularly the challenge of odd powers compared to even powers, which were resolved by Euler in the 18th century. The tools available today have advanced significantly, allowing for deeper exploration of these sums. While even powers have established solutions, odd powers remain elusive, with no general closed-form expression identified despite high-precision calculations. The harmonic series, which sums the reciprocals of natural numbers, diverges slowly, highlighting the complexity of these mathematical inquiries. Overall, the exploration of odd powers continues to be a significant area of research in mathematics.
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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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