JEE Advance - Mathematics (1985 - No. 18)
Find the sum of the series :
$$$\sum\limits_{r = 0}^n {{{\left( { - 1} \right)}^r}\,{}^n{C_r}\left[ {{1 \over {{2^r}}} + {{{3^r}} \over {{2^{2r}}}} + {{{7^r}} \over {{2^{3r}}}} + {{{{15}^r}} \over {{2^{4r}}}}..........up\,\,to\,\,m\,\,terms} \right]} $$$
{{{2^{mn}} + 1} \over {{2^{mn}}left( {{2^n} - 1}
ight)}}
{{{2^{mn}} - 1} \over {{2^{mn}}left( {{2^n} - 1}
ight)}}
{{{2^{m}} - 1} \over {{2^{n}}left( {{2^{mn}} - 1}
ight)}}
{{{2^{n}} - 1} \over {{2^{mn}}left( {{2^m} - 1}
ight)}}
{{{2^{mn}} - 1} \over {{2^{n}}left( {{2^{m}} - 1}
ight)}}
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