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

41. If \( \left\{a_{n}\right\} \) is a sequence of real numbers with \( \lim _{n \rightarrow \infty} \sup a_{n}=1 \), then

(a.) Set of values taken by \( \left\{a_{n}\right\} \) is not a finite set
(b.) \( \left\{a_{n}\right\} \) is convergent
(c.) \( \left\{a_{n}\right\} \) is bounded
(d.) \( \left\{a_{n}\right\} \) cannot diverge to \( +\infty \)

42. Let \( x_{1}=0, x_{2}=1 \), and for \( n \geq 3 \), define \( x_{n}=\frac{x_{n-1}+x_{n-2}}{2} \). Which of the following is/are true?

(a.) \( \left\{x_{n}\right\} \) is a monotone sequence.
(b.) \( \lim _{n \rightarrow \infty} x_{n}=\frac{1}{2} \).
(c.) \( \left\{x_{n}\right\} \) is a Cauchy sequence.
(d.) \( \lim _{n \rightarrow \infty} x_{n}=\frac{2}{3} \).

43. \( \lim _{n \rightarrow \infty} \frac{1}{\sqrt{n}}\left(\frac{1}{\sqrt{2}+\sqrt{4}}+\frac{1}{\sqrt{4}+\sqrt{6}}+\ldots \frac{1}{\sqrt{2 n}+\sqrt{2 n+2}}\right) \) is

(a.) \( \sqrt{2} \)
(b.) \( \frac{1}{\sqrt{2}} \)
(c.) \( \sqrt{2}+1 \)
(d.) \( \frac{1}{\sqrt{2}+1} \)

44. Let \( S_{n}=\sum_{k=1}^{n} \frac{1}{k} \). Which of the following is true?

(a.) \( S_{2^{n}} \geq \frac{n}{2} \) for every \( n \geq 1 \).
(b.) \( S_{n} \) is a bounded sequence.
(c.) \( \left|S_{2^{n}}-S_{2^{n-1}}\right| \rightarrow 0 \) as \( n \rightarrow \infty \).
(d.) \( \frac{S_{n}}{n} \rightarrow 1 \) as \( n \rightarrow \infty \).

45. \( L=\lim _{n \rightarrow \infty} \frac{1}{\sqrt[n]{n !}} \). Then

(a.) \( L=0 \)
(b.) \( L=1 \)
(c.) \( 0
(d.) \( L=\infty \)

46. Consider the sequence

\( a_{n}=\left(1+(-1)^{n} \frac{1}{n}\right)^{n} \)

Then

(a.) \( \limsup _{n \rightarrow \infty} a_{n}=\liminf _{n \rightarrow \infty} a_{n}=1 \)
(b.) \( \limsup _{n \rightarrow \infty} a_{n}=\liminf _{n \rightarrow \infty} a_{n}=e \)
(c.) \( \limsup _{n \rightarrow \infty} a_{n}=\liminf _{n \rightarrow \infty} a_{n}=\frac{1}{e} \)
(d.) \( \limsup _{n \rightarrow \infty} a_{n}=e, \liminf _{n \rightarrow \infty} a_{n}=\frac{1}{e} \)

47. Let \( \alpha=0.10110111011110 \ldots \) be a given real number written in base 10, that is, the \( n^{\text{th}} \) digit of \( \alpha \) is 1, unless \( n \) is of the form \( \frac{k(k+1)}{2}-1 \) in which case it is 0. Choose all the correct statements from below.

(a.) \( \alpha \) is a rational number
(b.) \( \alpha \) is an irrational number
(c.) For every integer \( q \geq 2 \), there exists an integer \( r \geq 1 \) such that \( \frac{r}{q}<\alpha<\frac{r+1}{q} \)
(d.) \( \alpha \) has no periodic decimal expansion

48. For \(a, b \in \mathbb{N}\), consider the sequence

\(d_{n}=\frac{\binom{n}{a}}{\binom{n}{b}}\)

for \(n>a, b\). Which of the following statements are true? As \(n \rightarrow \infty\)

(a.) \(\left\{d_{n}\right\}\) converges for all values of \(a\) and \(b\)
(b.) \(\left\{d_{n}\right\}\) converges if \(a < b\)
(c.) \(\left\{d_{n}\right\}\) converges if \(a=b\)
(d.) \(\left\{d_{n}\right\}\) converges if \(a>b\)

49. Let \(\left\{a_{n}\right\}\) be a sequence of real numbers satisfying \(\sum_{n=1}^{\infty}\left|a_{n}-a_{n-1}\right|<\infty\). Then the series \(\sum_{n=0}^{\infty} a_{n} x^{n}, x \in \mathbb{R}\) is convergent

(a.) nowhere on \(\mathbb{R}\)
(b.) everywhere on \(\mathbb{R}\)
(c.) on some set containing \((-1,1)\)
(d.) only on \((-1,1)\)

50. If \(\lambda_{n}=\int_{0}^{1} \frac{d t}{(1+t)^{n}}\) for \(n \in \mathbb{N}\), then

(a.) \(\lambda_{n}\) does not exist for some \(n\)
(b.) \(\lambda_{n}\) exists for every \(n\) and the sequence is unbounded
(c.) \(\lambda_{n}\) exists for every \(n\) and the sequence is bounded
(d.) \(\lim _{n \rightarrow \infty}\left(\lambda_{n}\right)^{1 / n}=1\)