A) \[log\text{ }2\]
B) \[\log (1+\sqrt{5})\]
C) \[log\text{ }6\]
D) 0
E) \[log\text{ }5\]
Correct Answer: C
Solution :
\[\underset{n\to \infty }{\mathop{\lim }}\,\left[ \frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{n+5n} \right]\] \[=\underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{r=0}^{5n}{\frac{1}{n+r}}=\underset{n\to \infty }{\mathop{\lim }}\,\left[ \frac{1}{n}\sum\limits_{r=0}^{5n}{\frac{n}{n+r}} \right]\] \[=\underset{n\to \infty }{\mathop{\lim }}\,\left[ \frac{1}{n}\sum\limits_{r=0}^{5n}{\frac{1}{1+(r/n)}} \right]\] \[=\int_{0}^{5}{\frac{1}{1+x}}dx=[\log (1+x)]_{0}^{5}\] \[=\log 6-\log 1=\log 6\]
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