calculation error in series

• To: mathgroup at smc.vnet.net
• Subject: [mg125641] calculation error in series
• From: Maurice Coderre <mauricecoderre at gmail.com>
• Date: Sat, 24 Mar 2012 02:02:57 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

```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?

```

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