A) \[\frac{1}{2}\log 2\]
B) \[\log \frac{3}{5}\]
C) \[\log \frac{5}{3}\]
D) \[\frac{1}{2}\log \frac{5}{3}\]
E) \[\frac{1}{2}\log \frac{3}{5}\]
Correct Answer: D
Solution :
\[\frac{1}{2}\left( \frac{1}{3}+\frac{1}{4} \right)-\frac{1}{4}\left( \frac{1}{{{3}^{2}}}+\frac{1}{{{4}^{2}}} \right)+\frac{1}{6}\left( \frac{1}{{{3}^{2}}}+\frac{1}{{{4}^{3}}} \right)-....\] \[=\frac{1}{2}\left[ \frac{1}{3}-\frac{1}{2}\left( \frac{1}{{{3}^{2}}} \right)+\frac{1}{3}\left( \frac{1}{{{3}^{3}}} \right)-.... \right]\] \[+\frac{1}{2}\left[ \frac{1}{4}-\frac{1}{2}\left( \frac{1}{{{4}^{2}}} \right)+\frac{1}{3}\left( \frac{1}{{{4}^{3}}} \right)-..... \right]\] \[=\frac{1}{2}\left[ \log \left( 1+\frac{1}{3} \right) \right]+\frac{1}{2}\left[ \log \left( 1+\frac{1}{4} \right) \right]\] \[=\frac{1}{2}\left[ \log \left( \frac{4}{3} \right)\times \left( \frac{5}{4} \right) \right]\] \[=\frac{1}{2}\log \left( \frac{5}{3} \right)\]
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