Continuous Function for this please

AI Thread Summary
The discussion revolves around finding a continuous function that generalizes the sum of terms like f = 1/a + 1/(a+1) + ... + 1/(a+(n-1)) for non-integer values of n. The user seeks a solution without interpolation, specifically using the digamma function to derive a closed form. They express familiarity with the Riemann zeta function but want a finite sum instead of one extending to infinity. The conversation highlights the use of harmonic numbers as a potential solution to the problem. Overall, the focus is on deriving a continuous function that meets these specific criteria.
TheDestroyer
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Continuous Function for this please !

Hi guys,

Whats the continuous function instead of this?

f = 1/a + 1/(a+1) + 1/(a+2) + 1/(a+3) + ... + 1/(a+(n-1))

or

f = 1/a + 1/(a-1) + 1/(a-2) + 1/(a-3) + ... + 1/(a-(n-1))


(a) is any number, and i want n to not be only an integer, i want it to be generalized as we did for (n-1)! = Gamma(n)

I know riemann zeta function but it takes the sum to the infinity, i want it only to a specific REAL number

Anyone can help about this? please, as i said in my previous post, no interpolation, and only continuous function is wanted,

and Thanks
 
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HOW?? Any one can help me with this?

The Equation (6) has a some to infinity ! can you guide me?
 
Thanks Any way, I fixed the problem using the harmonic numbers,
 

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