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calculation error in series


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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