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

11. Let \(p(x)\) be a polynomial in the real variable \(x\) of degree 5. Then \(\lim _{n \rightarrow \infty} \frac{p(n)}{2^{n}}\) is:

(a.) 5
(b.) 1
(c.) 0
(d.) \(\infty\)

12. If \(0 < c < d\), then the sequence \(a_{n}=\left(c^{n}+d^{n}\right)^{1/n}\) is:

(a.) bounded and monotonically decreasing
(b.) bounded and monotonically increasing
(c.) monotonically increasing, but unbounded for \(1 < c < d\)
(d.) monotonically decreasing, but unbounded for \(1 < c < d\)

13. The limit superior and the limit inferior of the following sequence \(\left\langle(-1)^{n}\left(1+\frac{1}{n}\right)\right\rangle\) are:

(a.) 1, -1
(b.) 2, -1
(c.) 2, 1
(d.) 1, 1

14. Let \(\left\{a_{n}\right\}\), \(\left\{b_{n}\right\}\), and \(\left\{c_{n}\right\}\) be sequences of real numbers such that \(b_{n}=a_{2n}\) and \(c_{n}=a_{2n+1}\). Then \(\left\{a_{n}\right\}\) being convergent:

(a.) implies \(\left\{b_{n}\right\}\) is convergent but \(\left\{c_{n}\right\}\) need not be convergent
(b.) implies \(\left\{c_{n}\right\}\) is convergent but \(\left\{b_{n}\right\}\) need not be convergent
(c.) implies both \(\left\{b_{n}\right\}\) and \(\left\{c_{n}\right\}\) are convergent
(d.) if both \(\left\{b_{n}\right\}\) and \(\left\{c_{n}\right\}\) are convergent

15. For each positive integer \(n\), let \(a_{n}\) be the number of points of intersection of the graph \(y=\sin x\) with the line \(y=\frac{x}{n}\). The sequence \(\left\{a_{n}\right\}\) is:

(a.) decreasing
(b.) constant
(c.) converging to zero
(d.) diverging to infinity

16. The value of \(\lim _{x \rightarrow \infty}\left(\frac{a_{1}^{1 / x}+a_{2}^{1 / x}+\ldots+a_{n}^{1 / x}}{n}\right)^{n x}\), \(a_{i}>0, \forall i\), is:

(a.) \(\frac{a_{1}+a_{2}+\ldots+a_{n}}{n}\)
(b.) \(e^{a_{1}+a_{2}+\ldots+a_{n}}\)
(c.) \(a_{1}+a_{2}+\ldots+a_{n}\)
(d.) \(a_{1} \cdot a_{2} \ldots . a_{n}\)

17. Let \(\left\langle x_{n}\right\rangle\) be a convergent sequence and \(\left\langle y_{n}\right\rangle\) be a monotonic sequence. Then \(\left\langle x_{n} \cdot y_{n}\right\rangle\):

(a.) Always converges
(b.) Converges if \(\left\langle x_{n}\right\rangle\) is monotonic
(c.) Converges if \(\left\langle x_{n} y_{n}\right\rangle\) is monotonic
(d.) Converges if \(\left\langle y_{n}\right\rangle\) is bounded

18. Consider the sequence \(\left\{l_{n}\right\}, n \in \mathbb{N}\) with \(l_{n}=\frac{1}{n+1}+\ldots+\frac{1}{2 n}\). This sequence:

(a.) is increasing and bounded
(b.) increases to \(\infty\)
(c.) decreases to 0
(d.) decreases to a positive number

19. The sequence \(\left\{S_{n}\right\}\), where \(S_{n}=1+\frac{1}{4}+\frac{1}{7}+\ldots+\frac{1}{3 n-2}\):

(a.) A Cauchy sequence
(b.) A convergent sequence
(c.) Cannot converge
(d.) None of these

20. The value of \(\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{\sqrt{n^{2}+k n}}\) is:

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