Practice Questions for NET JRF Real Analysis Assignment: Sequences of Real Numbers III

21. Consider the sequence \(\left\{a_{n}\right\}\) of real numbers where \(a_{1}>1\) and \(a_{n+1}=2-\frac{1}{a_{n}}\), \(n \geq 1\). Then the sequence \(\left\{a_{n}\right\}\) is:

(a.) bounded but not monotone
(b.) not bounded but monotone
(c.) both bounded and monotone
(d.) neither bounded nor monotone

22. Let \(\left\langle a_{n}\right\rangle\) be a sequence of real numbers define \(b_{n}\) and \(c_{n}\) as:

\(b_{n}\) = \(n\)th rational terms in \(\left\langle a_{n}\right\rangle\)

\(c_{n}\) = \(n\)th irrational terms in \(\left\langle a_{n}\right\rangle\)

Then:

(a.) \(\left\langle b_{n}\right\rangle\) and \(\left\langle c_{n}\right\rangle\) both are subsequences of \(\left\langle a_{n}\right\rangle\)
(b.) Both \(\left\langle b_{n}\right\rangle\) and \(\left\langle c_{n}\right\rangle\) may fail to be subsequences of \(\left\langle a_{n}\right\rangle\)
(c.) At least one of \(\left\langle b_{n}\right\rangle\) and \(\left\langle c_{n}\right\rangle\) is always a subsequence of \(\left\langle a_{n}\right\rangle\)
(d.) None of the above

23. The largest term in sequence \(x_{n}=\frac{1000^{n}}{n !}\), \(n=1,2,3,\ldots\), is:

(a.) \(x_{999}\)
(b.) \(x_{1001}\)
(c.) \(x_{1}\)
(d.) does not exist

24. Let the sequence \(\left\{x_{n}\right\}_{n \in \mathbb{N}}\) of real numbers converges to a non-zero real number \(a\) and let \(y_{n}=a-x_{n}\). Then \(\max _{n \in \mathbb{N}}\left\{x_{n}, y_{n}\right\}\) converges to:

(a.) \(a\) always
(b.) 0 always
(c.) \(\max \{a, 0\}\)

25. Let \(\left\{x_{n}\right\}\) be an unbounded sequence of nonzero real numbers. Then:

(a.) \(\left\{x_{n}\right\}\) must have a convergent subsequence
(b.) \(\left\{x_{n}\right\}\) cannot have a convergent subsequence
(c.) \(\left\{\frac{1}{x_{n}}\right\}\) must have a convergent subsequence
(d.) \(\left\{\frac{1}{x_{n}}\right\}\) cannot have a convergent subsequence

26. Let \(\left\{a_{n}\right\}\) be a sequence of real numbers. Let \(b_{n}=a_{n}+a_{n+1}\), for \(n=1,2,\ldots\). Which of the following is always true:

(a.) If \(\left\{b_{n}\right\}\) converges, then \(\left\{a_{n}\right\}\) converges
(b.) If \(\left\{a_{n}\right\}\) converges, then \(\left\{b_{n}\right\}\) diverges
(c.) If \(\left\{a_{n}\right\}\) converges, then \(\left\{b_{n}\right\}\) converges
(d.) \(\left\{a_{n}\right\}\) is a subsequence of \(\left\{b_{n}\right\}\)

27. \(x_{n+1}=\frac{-3}{4} x_{n}\), \(x_{0}=1\). The sequence \(\left\{x_{n}\right\}\):

(a.) diverges
(b.) \(x_{n}\) is monotonically increasing and converges to 0
(c.) \(x_{n}\) is monotonically decreasing and converges to 0
(d.) none of the above

28. Let \(\left\{x_{n}\right\}\) be a sequence of real numbers such that \(\lim _{n \rightarrow \infty}\left(x_{n+1}-x_{n}\right)=c\), where \(c\) is a positive real number. Then the sequence \(\left\{\frac{x_{n}}{n}\right\}\):

(a.) is NOT bounded
(b.) is bounded but NOT convergent
(c.) converges to \(c\)
(d.) converges to 0

29. If \(\left\langle a_{n}\right\rangle\) is a convergent sequence then \(\lim _{n \rightarrow \infty} n\left(a_{n+1}-a_{n}\right)\):

(a.) \(\infty\)
(b.) 1
(c.) \(e\)
(d.) none of the above

30. Suppose \(a>0\). Consider the sequence \(a_{n}=n\left(\sqrt[n]{e a}-\sqrt[n]{a}\right)\), \(n \geq 1\). Then:

(a.) \(\lim _{n \rightarrow \infty} a_{n}\) does not exist
(b.) \(\lim _{n \rightarrow \infty} a_{n}=e\)
(c.) \(\lim _{n \rightarrow \infty} a_{n}=0\)
(d.) None of the above