A) \[\left( \frac{n}{n+1} \right)\]
B) \[{{\left( \frac{n}{n+1} \right)}^{2}}\]
C) \[{{\left( \frac{n}{n+1} \right)}^{4}}\]
D) \[{{\left( \frac{n}{n+1} \right)}^{6}}\]
Correct Answer: B
Solution :
\[\sum\limits_{k=1}^{n}{{{a}_{k}}}=\sum\limits_{k=1}^{n}{\frac{1}{k\,(k+1)}}\] =\[\left( 1-\frac{1}{2} \right)+\left( \frac{1}{2}-\frac{1}{3} \right)+...+\left( \frac{1}{n}-\frac{1}{n+1} \right)\] = \[1-\frac{1}{n+1}=\frac{n}{n+1}\] \[{{\left( \sum\limits_{k=1}^{n}{{{a}_{k}}} \right)}^{2}}={{\left( \frac{n}{n+1} \right)}^{2}}\].
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