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

31. \(\lim _{n \rightarrow \infty}\left[\left(\frac{2}{1}\right)\left(\frac{3}{2}\right)^{2}\left(\frac{4}{3}\right)^{3} \ldots \ldots \ldots .\left(\frac{n+1}{n}\right)^{n}\right]^{1 / n}=\)

(a.) 1
(b.) -1
(c.) \(-e\)
(d.) \(e\)

32. Which of the following is/are correct?

(a.) \(n \log \left(1+\frac{1}{n+1}\right) \rightarrow 1\) as \(n \rightarrow \infty\)
(b.) \((n+1) \log \left(1+\frac{1}{n+1}\right) \rightarrow 1\) as \(n \rightarrow \infty\)
(c.) \(n^{2} \log \left(1+\frac{1}{n}\right) \rightarrow 1\) as \(n \rightarrow \infty\)
(d.) \(n \log \left(1+\frac{1}{n^{2}}\right) \rightarrow 1\) as \(n \rightarrow \infty\)

33. If \(\left\{x_{n}\right\}\) and \(\left\{y_{n}\right\}\) are sequences of real numbers, which of the following is/are true?

(a.) \(\limsup _{n}\left(x_{n}+y_{n}\right) \leq \lim \sup _{n}+\limsup _{n} y_{n}\)
(b.) \(\limsup _{n}\left(x_{n}+y_{n}\right) \geq \limsup _{n} x_{n}+\limsup _{n} y_{n}\)
(c.) \(\liminf _{n}\left(x_{n}+y_{n}\right) \leq \liminf _{n} x_{n}+\liminf _{n} y_{n}\)
(d.) \(\liminf _{n}\left(x_{n}+y_{n}\right) \geq \liminf _{n} x_{n}+\liminf _{n} y_{n}\)

34. If \(\left\langle a_{n}\right\rangle\) is a sequence of real numbers such that for some \(l \in \mathbb{R}\), define a set \(K=\left\{n \in \mathbb{N}: a_{n} \notin (l-\varepsilon, l+\varepsilon), \forall \varepsilon>0\right\}\) if \(K\) is bounded, then choose the incorrect:

(a.) \(l\) is a limit point of \(\left\langle a_{n}\right\rangle\) but \(\left\langle a_{n}\right\rangle\) may fail to be convergent
(b.) \(\left\langle a_{n}\right\rangle\) is monotonic.
(c.) \(\left\langle a_{n}\right\rangle\) is a Cauchy sequence
(d.) \(\left\langle a_{n}\right\rangle\) is bounded but need not be convergent

35. Let \(\left\langle a_{n}\right\rangle\) and \(\left\langle b_{n}\right\rangle\) be two sequences. Which of the following is incorrect?

(a.) If \(\left\langle a_{n}+b_{n}\right\rangle\) and \(\left\langle a_{n} \cdot b_{n}\right\rangle\) are both convergent, then \(\left\langle a_{n}\right\rangle\) and \(\left\langle b_{n}\right\rangle\) are convergent
(b.) If \(\left\langle a_{n}-b_{n}\right\rangle\) and \(\left\langle\frac{a_{n}}{b_{n}}\right\rangle\) are both convergent, then \(\left\langle a_{n}\right\rangle\) and \(\left\langle b_{n}\right\rangle\) are convergent
(c.) If \(\left\langle a_{n}+b_{n}\right\rangle\) and \(\left\langle a_{n}-b_{n}\right\rangle\) are both convergent, then \(\left\langle a_{n}\right\rangle\) and \(\left\langle b_{n}\right\rangle\) are convergent
(d.) If \(\left\langle a_{n}+b_{n}\right\rangle\) and \(\left\langle a_{n}-b_{n}\right\rangle\) are both convergent, then \(\left\langle a_{n}. b_{n}\right\rangle\) is convergent

36. Which of the following are null sequences?

(a.) \(\left\{\frac{n}{\alpha^{n}}\right\}: \alpha>1\)
(b.) \(\left\{\frac{\log n}{n}\right\}\)
(c.) \(\left\{\alpha^{n}\right\}: |\alpha|<1\)
(d.) \(\left(1+\frac{3}{n}\right)^{n}: n \in \mathbb{N}\)

37. Which of the following is/are incorrect statements?

(a.) Every sequence contains a monotonic subsequence
(b.) Every limit point of the sequence is the limit of the range set of the sequence
(c.) If a sequence has a unique limit point, then it must be the limit of the sequence
(d.) The set of disjoint open intervals in \(\mathbb{R}\) is an uncountable set

38. Let \(\left\{x_{n}\right\}\) and \(\left\{y_{n}\right\}\) be two sequences of real numbers such that \(x_{n} \leq y_{n} \leq x_{n+2}\), \(n=1,2,3, \ldots\).

(a.) \(\left\{y_{n}\right\}\) is a bounded sequence
(b.) \(\left\{x_{n}\right\}\) is an increasing sequence
(c.) \(\left\{x_{n}\right\}\) and \(\left\{y_{n}\right\}\) converge together
(d.) \(\left\{y_{n}\right\}\) is an increasing sequence

39. Let \( \left\langle a_n \right\rangle \) be a sequence of real numbers defined as \( a_n = \begin{cases} -1 - \frac{1}{n} & n \text { is even } \\ 1 + \frac{1}{n} & n \text { is odd } \end{cases} \).

Then which of the following is correct:

(a.) \( \left\langle a_n \right\rangle \) is oscillatory
(b.) If \( \lim \inf \left\langle a_n \right\rangle = a \) and \( \lim \sup \left\langle a_n \right\rangle = b \), then \( a_n \notin (a, b) \) for \( \forall n \in \mathbf{N} \)
(c.) \( S = \left\{ \lim a_{n_k}:\left\langle a_{n_k} \right\rangle\right. \) is a convergent subsequence of \( \left. \left\langle a_n \right\rangle \right\} \) is finite
(d.) None of these

40. Which of the following is not possible about \( \left\langle a_{n} \right\rangle \)

(a.) \( \sum a_{n} \) is convergent but \( \left\langle a_{n} \right\rangle \) is non-convergent
(b.) \( \left\langle \left|a_{n} \right| \right\rangle \) is convergent but \( \left\langle a_{n} \right\rangle \) is non-convergent
(c.) \( \left\langle a_{n} \right\rangle \) is convergent but \( \left\langle \left|a_{n} \right| \right\rangle \) is non-convergent
(d.) None of these