Sum[(1/(k (k + 1))) Sum[j DigitCount[k, b, j], {j, 1, b - 1}], {k, 1, Infinity}] == (b Log[b])/(b - 1) /; Element[b - 1, Integers] && b - 1 > 0

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mi> j </mi> <mo> &#8290; </mo> <mrow> <msubsup> <mi> s </mi> <mi> b </mi> <mrow> <mo> ( </mo> <mi> j </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <ci> k </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <ci> j </ci> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> s </ci> <ci> b </ci> </apply> <ci> j </ci> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <ln /> <ci> b </ci> </apply> <apply> <power /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29