Sum[Abs[Fibonacci[k] GoldenRatio - Fibonacci[k + 1]], {k, 1, Infinity}] == GoldenRatio

Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List["Abs", "[", RowBox[List[RowBox[List[RowBox[List["Fibonacci", "[", "k", "]"]], " ", "GoldenRatio"]], "-", RowBox[List["Fibonacci", "[", RowBox[List["k", "+", "1"]], "]"]]]], "]"]]]], "\[Equal]", "GoldenRatio"]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mi> k </mi> </msub> <mo> &#8290; </mo> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, Function[List[], GoldenRatio]] </annotation> </semantics> </mrow> <mo> - </mo> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> &#63449; </mo> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, Function[List[], GoldenRatio]] </annotation> </semantics> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <abs /> <apply> <plus /> <apply> <times /> <apply> <ci> Fibonacci </ci> <ci> k </ci> </apply> <ci> GoldenRatio </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Fibonacci </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <ci> GoldenRatio </ci> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "1"]], "\[Infinity]"], RowBox[List["Abs", "[", RowBox[List[RowBox[List[RowBox[List["Fibonacci", "[", "k_", "]"]], " ", "GoldenRatio"]], "-", RowBox[List["Fibonacci", "[", RowBox[List["k_", "+", "1"]], "]"]]]], "]"]]]], "]"]], "\[RuleDelayed]", "GoldenRatio"]]]]

2007-05-02