Question

Let $a_n={log}_n(n+1),n\ge 2$ is an integer, mark the incorrect statement :

• A. $\underset{n\to \infty }{lim}{a}_n=e$
• B. Sequence ${(a}_n)$ has no greatest term.
• C. ${log}_{6}7<{log}_{7}8$
• D. The greatest term of ${(a}_n)$ is ${log}_{2}3$.

Answer 1

The correct option is A $\underset{n\to \infty }{lim}{a}_n=e$

Given,

$a_n={\log }_n(n+1)$

${lim}_{n\to \infty }{a}_n$

$={lim}_{n\to \infty }\left({\log }_n(n+1)\right)$

${lim}_{n\to \infty }(n+1)=\infty$

${lim}_{u\to \infty }\left({\log }_u\left(u\right)\right)=1$

$\therefore {lim}_{n\to \infty }{a}_n=1$