Is there any general solution exist for $\sum_{i=1}^{N} \frac{1}{B(x+i, i)}$ function?

Question

$\sum_{i=1}^{N} \frac{1}{B(x+i, i)} = \frac{1}{B(x+1, 1)} + \frac{1}{B(x+2, 2)} + \ldots + \frac{1}{B(x+N, N)}$

where ${N}$ is fixed value. is there any simple expression exist for x in the above expression?

or x converges to somewhere if ${N}$ is enoughly large?

Welcome to the Mathematica Stack Exchange. As written, this question can be more suitably answered at the MathSE site. If this is a Mathematica related question, then please include code that you have tried so far and include a definition for B. Also note that N is a reserved symbol in Mathematica. Thanks. — Syed, Sep 15 '23 at 09:03

Bob Hanlon · Accepted Answer · 2023-09-15T20:55:19.033

4

$Version

(* "13.3.1 for Mac OS X ARM (64-bit) (July 24, 2023)" *)

Clear["Global`*"]

Assuming that B represents the Euler beta function

(sum[n_, x_] = Inactive[Sum][1/Beta[x + i, i], {i, 1, n}]) // 
  TraditionalForm

This sum is

(sum[n_, x_] = sum[n, x] // Activate // FullSimplify) // 
  TraditionalForm

The infinite sum diverges; however, it can be Borel regularized

(sumReg[x_] = Sum[1/Beta[x + i, i], {i, 1, ∞},
    Regularization -> "Borel"]) // TraditionalForm

EDIT: As noted by Roman, sumReg[x] can be simplified

(sumReg[x_] = sumReg[x] // FullSimplify) // 
  TraditionalForm

This function has odd symmetry

sumReg[-x] == -sumReg[x] // FullSimplify
(* True *)
Plot[sumReg[x], {x, 0, 10 π},
 Frame -> True,
 FrameLabel -> (Style[#, 14] & /@ {x, sumReg})]

edited Sep 15 '23 at 20:55

answered Sep 15 '23 at 19:28

Bob Hanlon

Using FullSimplify on the Borel-regularized result gives a much simpler $\frac{1}{9} \left(\sqrt{3} \sin \left(\frac{\pi x}{3}\right)-3 x \cos \left(\frac{\pi x}{3}\right)\right)$. – Roman Sep 15 '23 at 19:55
@Roman - thanks, I added that. – Bob Hanlon Sep 15 '23 at 20:55
Instead of bitmap images of TraditionalForm expressions, you can use TeXForm and paste the result here for automatic rendering. – Roman Sep 16 '23 at 05:10

1 Answers1