Novel Constant \sum and \int s^{-s} - Defining a Fundamental Constant?

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SUMMARY

The summation \(\sum_{s=1}^{\infty}s^{-s}\) evaluates to approximately 1.291285997062663540407282590595600541498619368274, while the integral \(\int_{1}^{\infty}s^{-s}ds\) yields approximately 1.99545595750013800041872469845272435208621663. Both calculations are straightforward and can be computed to a high degree of precision. Notably, the integral from 0 to 1 matches the value of the summation, but no significant properties were identified for the integral from 1 to infinity. These results do not appear to derive directly from the mathematical constant e.

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Loren Booda
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\sum _{s =1}^{\infty}s^{-s}= ?

\int _{1}^{\infty}s^{-s}ds= ?

Does either this summation or integral define a fundamental constant?
 
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The sum is 1.291285997062663540407282590595600541498619368274..., Sloane's http://www.research.att.com/~njas/sequences/A073009 .

The integral is 1.99545595750013800041872469845272435208621663... Both sum and integral are easy to calculate to a large number of decimal places.

The integral over 0 to 1 has the same value as the sum. I wasn't able to find anything interesting about the integral from 1 to infinity.
 
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Thank you for your research, CR. I guess they have no direct derivation from the constant e.
 

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