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

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  • #1
Loren Booda
3,125
4
[tex] \sum _{s =1}^{\infty}s^{-s}= ? [/tex]

[tex] \int _{1}^{\infty}s^{-s}ds= ? [/tex]

Does either this summation or integral define a fundamental constant?
 
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  • #2
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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  • #3
Thank you for your research, CR. I guess they have no direct derivation from the constant e.
 

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

1. What is the concept of a "Novel Constant"?

A "Novel Constant" is a mathematical constant that is newly discovered or defined. It is typically denoted by a letter or symbol and has a specific value that remains constant in mathematical equations and formulas.

2. What does the symbol \sum represent in "Novel Constant \sum and \int s^{-s} - Defining a Fundamental Constant"?

The symbol \sum represents the summation operator, which is commonly used in mathematics to represent the sum of a series of terms. In this context, it is used to represent the sum of the values of the novel constant, \int s^{-s}.

3. How is the value of a novel constant determined?

The value of a novel constant is determined through mathematical calculations or experiments. It is often defined in terms of other known constants or values and can also be derived from fundamental principles and equations.

4. What is the purpose of defining a Fundamental Constant?

A fundamental constant is a fixed value that plays a crucial role in mathematical equations and formulas. By defining a fundamental constant, it allows for a more concise and consistent representation of these equations and simplifies complex calculations. It also provides a universal standard for measurements and comparisons in different fields of study.

5. How does the novel constant \int s^{-s} relate to other well-known mathematical constants?

The novel constant \int s^{-s} is not directly related to other well-known mathematical constants. However, it may be used in conjunction with other constants to define new mathematical concepts and principles. It could also potentially be used in the development of new mathematical formulas and equations.

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