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Answer
Correct option: B.$\frac{1}{2}\,$
b
$\int\limits_{\ln 2}^{\ln 3} {} (e^{-x} + 2 e^{-2x} + 3 e^{- 3x} +...... + \infty ) dx$ $= - \left[ {{e^{ - x}}\,\, + \,\,{e^{ - 2x}}\,\, + \,{e^{ - 3x}}\, + {e^{ - 4x}}\, + \,......} \right]_{\ln 2}^{\ln 3}$
$= \left[ {\left\{ {\frac{1}{3}\, + \,\frac{1}{{{3^2}}}\,\, + \,\frac{1}{{{3^3}}}\,\, + .......} \right\}\,\,\, - \,\,\,\left\{ {\frac{1}{2}\, + \,\frac{1}{{{2^2}}}\,\, + \,\frac{1}{{{2^3}}}\,\, + .......} \right\}} \right]$
$= \frac{{1/2}}{{1 - 1/2}}\,\, = \,\,\frac{{1/3}}{{1 - 1/3}}\,\, = \,\,1 - \frac{1}{2}\,\, = \,\,\frac{1}{2}\,\,$
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