TIFR ECE 2012 | Question: 3

Question

A sequence of numbers $\left(x_{n}: n=1,2,3, \ldots\right)$ is said to have a limit $x$, if given any number $\epsilon>0$, there exists an integer $n_{\epsilon}$ such that
\[
\left|x_{n}-x\right|<\epsilon
\]
for all $n \geq n_{\epsilon}$. In other words a sequence $\left(x_{n}: n=1,2,3, \ldots\right)$ has a limit $x$ all but a finite number of points of this sequence are arbitrarily close to $x$. Now consider a sequence
\[
x_{n}=5+\frac{(-1)^{n}}{n}+\left(1-\frac{271}{2^{n}}\right)
\]
for all $n \geq 1$. Which of the following statements is true?

• A. The sequence $\left(x_{n}: n=1,2,3, \ldots\right)$ fluctuates around $6$ and has a limit that equals $6 .$
• B. The sequence $\left(x_{n}: n=1,2,3, \ldots\right)$ oscillates around $5$ and does not have a limit.
• C. The sequence $\left(x_{n}: n=1,2,3, \ldots\right)$ oscillates around $6$ and does not have a limit.
• D. The sequence $\left(x_{n}: n=1,2,3, \ldots\right)$ is eventually always below $6$ and has a limit that equals $6$ .
• E. None of the above statements are true.