Question

Show that

$\sf log_{e} \sqrt{12} = 1 + \Bigg( \dfrac{ 1}{2} + \dfrac{1}{3}\Bigg) \dfrac{1}{4} + \Bigg( \dfrac{ 1}{4} + \dfrac{1}{5} \Bigg) \dfrac{1}{ {4}^{2} } + \Bigg( \dfrac{1}{6} + \dfrac{1}{7} \Bigg ) \dfrac{1}{ {4}^{3} } + ....$
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Answer 1

$\rm{Consider \: the \: R.H.S = \: \sf 1 + \Bigg( \dfrac{ 1}{2} + \dfrac{1}{3}\Bigg) \dfrac{1}{4} + \Bigg( \dfrac{ 1}{4} + \dfrac{1}{5} \Bigg) \dfrac{1}{ {4}^{2} } + \Bigg( \dfrac{1}{6} + \dfrac{1}{7} \Bigg ) \dfrac{1}{ {4}^{3} } + .... }$

$\sf = \Bigg( \dfrac{1}{2}. \dfrac{1}{4} + \dfrac{1}{4} . \dfrac{1}{ {4}^{2} } + \dfrac{1}{6} . \dfrac{1}{ {4}^{3} } + ... \Bigg ) + \Bigg( 1 + \dfrac{1}{3}. \dfrac{1}{4} + \dfrac{1}{5}. \dfrac{1}{ {4}^{2} } + \dfrac{1}{7} . \dfrac{1}{ {4}^{3} } + ... \Bigg )$

$\sf \: = \Bigg [ \dfrac{1}{2} .\bigg( { \dfrac{1}{2}\bigg) }^{2} + \dfrac{1}{4} \bigg( { \dfrac{1}{2}\bigg ) }^{4} + \dfrac{1}{6} \bigg( { \dfrac{1}{2}\bigg ) }^{6} + ...\Bigg ] + \Bigg [ 1 + \dfrac{1}{3} .\bigg( { \dfrac{1}{2}\bigg) }^{2} + \dfrac{1}{5} \bigg( { \dfrac{1}{2}\bigg ) }^{4} + \dfrac{1}{7} \bigg( { \dfrac{1}{2}\bigg ) }^{6} + ...\Bigg ]$

$\sf = \Bigg[ \dfrac{1}{2} {x}^{2} + \dfrac{1}{4} {x}^{4} + \dfrac{1}{6} {x}^{6} + ...\Bigg ] + \Bigg[ 1 + \dfrac{1}{3} {x}^{2} + \dfrac{1}{5} {x}^{4} + \dfrac{1}{6} {x}^{7} + ...\Bigg ] \: \: \: \: \: \: \: \bigg(taking \: x = \dfrac{1}{2} \bigg)$

$\sf = \dfrac{1}{2} \bigg \{ {x}^{2} + \dfrac{1}{2} {x}^{4} + \dfrac{1}{3} {x}^{6} + ... \bigg \} + \sf \dfrac{1}{x} \bigg \{ x + \dfrac{1}{3} {x}^{3} + \dfrac{1}{5} {x}^{5} + ... \bigg \}$

$\sf = - \dfrac{1}{2} log_{e}(1 - {x}^{2} ) + \dfrac{1}{2x} log_{e} \bigg( \dfrac{1 + x}{1 - x} \bigg)$

$\sf \: = - \dfrac{1}{2} log_{e} \bigg(1 - \dfrac{1}{4} \bigg) + log_{e}\Bigg ( \dfrac{ 1 + \dfrac{1}{2} }{1 - \dfrac{1}{2} } \Bigg )$

$\sf \: = \: - \dfrac{1}{2} log_{e} \bigg( \dfrac{3}{4} \bigg) + log_{e}(3) = - \dfrac{1}{2} log_{e} \bigg( \dfrac{3}{4} \bigg ) + \dfrac{1}{2} log_{e}(9)$

$\sf= \dfrac{1}{2} log_{e} \bigg( \dfrac{9 \times 4}{3} \bigg)$

$\sf \: = \dfrac{ log_{e}(12) }{2}$

$\sf \boxed{ \boxed{ \sf = log_{e}( \sqrt{12} ) }}$