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 Exp

 http://functions.wolfram.com/01.03.23.0002.01

 Input Form

 Sum[(-1)^(k - 1)/E^(k z), {k, 1, Infinity}] == 1/(E^z + 1) /; Re[z] > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "k"]], " ", "z"]]]]]]], "\[Equal]", FractionBox["1", RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], "+", "1"]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]

 MathML Form

 k = 1 ( - 1 ) k - 1 - k z 1 z + 1 /; Re ( z ) > 0 Condition k 1 -1 k -1 -1 k z 1 z 1 -1 z 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "1"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k_", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "k_"]], " ", "z_"]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], "+", "1"]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2001-10-29