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Re: calculation error in series

  • To: mathgroup at smc.vnet.net
  • Subject: [mg125648] Re: calculation error in series
  • From: Peter Pein <petsie at dordos.net>
  • Date: Sun, 25 Mar 2012 00:16:23 -0500 (EST)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <jkjrlc$435$1@smc.vnet.net>

Am 24.03.2012 08:04, schrieb Maurice Coderre:
> In[52]:= \!\(
> \*UnderoverscriptBox[\(\[Sum]\), \(n = 0\), \(\[Infinity]\)]\(
> FractionBox[\(1\),
> SuperscriptBox[\(2\), \((n + 1)\)]] \(
> \*UnderoverscriptBox[\(\[Sum]\), \(k = 0\), \(n\)]\((
> SuperscriptBox[\((\(-1\))\), \(k\)] \((\((
> \*FractionBox[\(n!\), \(\(\((n - k)\)!\) \(k!\)\)])\)
> \*SuperscriptBox[\(E\), \(-
> \*FractionBox[\(k\), \(2\)]\)]\ )\) Cos[14.134725141734695  k])\)\)\)
> \)
>
> Out[52]= 0.730559318177 + 5.55111512313*10^-17 I
>
> In[53]:= \!\(
> \*UnderoverscriptBox[\(\[Sum]\), \(n = 0\), \(\[Infinity]\)]\(
> FractionBox[\(1\),
> SuperscriptBox[\(2\), \((n + 1)\)]] \(
> \*UnderoverscriptBox[\(\[Sum]\), \(k = 0\), \(n\)]\((
> SuperscriptBox[\((\(-1\))\), \(k\)] \((\((
> \*FractionBox[\(n!\), \(\(\((n - k)\)!\) \(k!\)\)])\)
> \*SuperscriptBox[\(E\), \(-
> \*FractionBox[\(k\), \(2\)]\)]\ )\))\)\)\)\)
>
> Out[53]= Sqrt[E]/(1 + Sqrt[E])
>
> Why does the insertion of a purely real trigonometric function in a
> purely real infinit series, as shown above, give a complex result? Is
> it the result of an accumulated imprecision in the numerical
> evaluation?
>

Hi Maurice,

one can not expect exact results from inexact input. You can get rid of 
the floatingpoint-noise by applying Chop to the result or - preferably - 
by calculating the sum for an exact constant and replacing the 
approximate value later:

In[3]:= Sum[(1/2^(n + 1))*
           Sum[(-1)^k*(n!/E^(k/2))*Cos[c*k]/
                ((n - k)!*k!), {k, 0, n}],
          {n, 0, Infinity}]//ExpToTrig//FullSimplify
Out[3]= (E+Sqrt[E] Cos[c])/(1+E+2 Sqrt[E] Cos[c])
In[4]:= %/.c->14.134725141734695
Out[4]= 0.730559

Peter

P.S.: please use "Copy as Plain Text" (Shift-Strg-C) to copy/paste parts 
of your notebooks. Thanks



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