Re: Can it be done - easily?

• To: mathgroup at smc.vnet.net
• Subject: [mg13241] Re: [mg13211] Can it be done - easily?
• From: Robert Pratt <rpratt at math.unc.edu>
• Date: Fri, 17 Jul 1998 03:17:38 -0400
• Sender: owner-wri-mathgroup at wolfram.com

Whether you like this or not will depend on your definition of  "simple"
formula.  Paste the following into a notebook to view it.   Note that
the x can just be factored out of the sum.

z[x_,a_,b_,k_]:=Sum[x/y,{y,a,b,k}]

z[10000,100,1000,100]

\!\(36905\/126\)

N[%]

292.897

z[x,a,b,k]

\!\(\(x\ \((
\(-PolyGamma[0, a\/k]\) +
PolyGamma[0, 1 + a\/k + Floor[\(\(-a\) + b\)\/k]])\)\)\/k\)

?PolyGamma

"PolyGamma[z] gives the digamma function psi(z). PolyGamma[n, z]  gives
the \
nth derivative of the digamma function."

Rob Pratt
Department of Mathematics
The University of North Carolina at Chapel Hill CB# 3250, 331 Phillips
Hall
Chapel Hill, NC  27599-3250

rpratt at math.unc.edu

On Mon, 13 Jul 1998, Barry Culhane wrote:

> Myself and two workmates are software developers.  One guy wanted a
> formula to calculate a result for the following equation...
>      Z = sum of X/Y where X is a fixed number, and Y ranges from A-B in
> fixed steps...
>      i.e... X=10000 ; Y=100,200,300...1000
>      i.e... Z= 10000/100 + 10000/200 + ... 10000/1000 =  292.896
>
> He and I tried to figure out a simple formula to calculate it, but
> couldn't. The third guy said it was *not* *possible* to derive a
> formula - we think he's wrong, but can't prove it.  MathCad can solve
> it in the blink of an eye, even if the value of Y ranges from 1 to 1e6
> in steps of 1 !!!
>
> Can anyone come up with a simple formula to give a reasonably accurate
> result?  It is too slow to actually divide X by Y for each value of Y
> as there may be 1000 or even 100,000 values of Y.
>