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Can someone explain why these outputs differ?

In[5]:= g1 = 
   Integrate[Exp[-I x (n + 1)]/( Exp[I x] + 2), {x, 0, 2 Pi}, 
   Assumptions -> Element[n, Integers]];

In[6]:= g1 /. n -> 3

During evaluation of In[6]:= Infinity::indet: Indeterminate expression 0 ComplexInfinity encountered.

Out[6]= Indeterminate

In[7]:= g2 = Integrate[Exp[-I x (3 + 1)]/( Exp[I x] + 2), {x, 0, 2 Pi}]

Out[7]= π/16

The last is correct, the first seems to apply some incorrect assumptions? Note

In[8]:= g1

Out[8]= 2^(-1 - n) Beta[-(1/2), -1 - n, 0] Sin[n π]

See Background for context.

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  • 1
    $\begingroup$ General result is valid: {Block[{$MaxExtraPrecision = 300, eps = 10^-300}, N[Limit[2^(-1 - n) Beta[-(1/2), -1 - n, 0] Sin[n \[Pi]], n -> 3 + eps], 100]] // Chop, N[Pi/16, 100]} $\endgroup$ – Mariusz Iwaniuk Jul 14 '18 at 14:37
  • $\begingroup$ Thanks you @MariuszIwaniuk, that answers it for me. $\endgroup$ – Sooner Jul 18 '18 at 10:33
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Here is a workaround

g1[n_?IntegerQ] := Integrate[Exp[-I x (n + 1)]/(Exp[I x] + 2), {x, 0, 2 Pi} ];    
{g1[1], g1[2], g1[3] }
{π/4, -(π/8), π/16}
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