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MathGroup Archive 1998

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Re: Can it be done - easily?

  • To: mathgroup at
  • Subject: [mg13236] Re: Can it be done - easily?
  • From: Daniel Lichtblau <danl>
  • Date: Fri, 17 Jul 1998 03:17:35 -0400
  • Organization: Wolfram Research, Inc.
  • References: <6od25q$>
  • Sender: owner-wri-mathgroup at

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.
> Thanks in advance...
> > Barry Culhane
> > Schaffner Ltd, Limerick, IRELAND

One method:

In[20]:= Timing[InputForm[ee = Sum[1/y, {y,a,b,c}]]]
Out[20]= {0.32 Second, -(PolyGamma[0, a/c]/c) +
>     PolyGamma[0, 1 + a/c + Floor[(-a + b)/c]]/c}

In[21]:= InputForm[ff = x*ee /. {x->10000, a->100, b->1000, c->100}]
Out[21]//InputForm= 10000*((7381/2520 - EulerGamma)/100 +

In[22]:= N[ff]
Out[22]= 292.897

You may need to use high precision rather than machine arithmetic in the
last step. Depends on the magnitudes of the numbers in the exact

Daniel Lichtblau
Wolfram Research

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