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

11. Consider the series \(\sum \frac{1}{\sqrt{n}}\) and \(\sum \frac{1}{n^{3 / 2}}\). Then

(a.) both the series converge to the same value
(b.) both the series converge to different values
(c.) both the series are divergent
(d.) the first series is divergent and the second series is convergent

12. If the terms of this oscillating series are grouped pairwise \(3-\frac{2^{2}+1}{2}+\frac{2^{3}+1}{2^{2}}-\frac{2^{4}+1}{2^{3}}+\ldots\), then the resulting series becomes

(a.) Convergent
(b.) Divergent
(c.) Oscillate finitely
(d.) Oscillate infinitely

13. Which of the following series is divergent?

(a.) \(\sum_{n=1}^{\infty} \sqrt{n} \sin \frac{1}{n}\)
(b.) \(\sum_{n=1}^{\infty} \sqrt{n} \sin ^{-1}\left(\frac{1}{n^{2}}\right)\)
(c.) \(\sum_{n=1}^{\infty} n \tan \left(\frac{1}{n^{3}}\right)\)
(d.) \(\sum_{n=1}^{\infty} \tan ^{-1}\left(\frac{1}{n^{2}}\right)\)

14. The largest interval in which the series \(\sum_{n=1}^{\infty} x^{n}\) converges

(a.) For \(x\) with \(-1<x<1\)
(b.) For \(x\) with \(-\frac{1}{2}<x<\frac{1}{2}\)
(c.) For \(x\) with \(-2<x<2\)
(d.) For \(x\) with \(-\frac{1}{2} \leq x \leq \frac{1}{2}\)

15. The series \(\left(\frac{2^{2}}{1^{2}}-\frac{2}{1}\right)^{-1}+\left(\frac{3^{3}}{2^{3}}-\frac{3}{2}\right)^{-2}+\left(\frac{4^{4}}{3^{4}}-\frac{4}{3}\right)^{-3}+\ldots\) is

(a.) Convergent
(b.) Divergent
(c.) Oscillate finitely
(d.) Oscillate infinitely

16. The series \(x+\frac{2^{2} x^{2}}{2 !}+\frac{3^{3} x^{3}}{3 !}+\frac{4^{4} x^{4}}{4 !}+\ldots\) is convergent if

(a.) \(0
(b.) \(x>\frac{1}{e}\)
(c.) \(\frac{2}{e}
(d.) \(\frac{3}{e}

17. Consider the following statements:

1. If \(\sum_{n=1}^{\infty} u_{n}\) is a series of positive terms, then the convergence of \(\sum_{n=1}^{\infty}(-1)^{n} u_{n}\) implies the convergence of \(\sum_{n=1}^{\infty} u_{n}\).

2. If \(\sum_{n=1}^{\infty} u_{n}\) is a series of positive terms then the convergence of \(\sum_{n=1}^{\infty} u_{n}\) implies the convergence of \(\sum_{n=1}^{\infty}(-1)^{n} u_{n}\).

3. The convergence of \(\sum_{n=1}^{\infty} u_{n}\) \(\left(u_{n}>0\right)\) implies the convergence of \(\sum_{n=1}^{\infty} u_{n}^{2}\).

Select the correct answer using the codes given below

(a.) 2 and 3
(b.) 1 and 3
(c.) 1 and 2
(d.) 1, 2 and 3

18. Let the series \(\sum_{n=1}^{\infty} u_{n}\) has bounded partial series, then the series \(\sum_{n=1}^{\infty} u_{n} v_{n}\) is convergent if the sequence \(\left\{v_{n}\right\}\) is

(a.) Monotonic and bounded
(b.) Monotonic and converges to zero
(c.) Bounded and converges to zero
(d.) Bounded and monotonic which converges to zero

19. If \(p\) is a real number, then the series \(\frac{1}{1^{p}}+\frac{1}{3^{p}}+\frac{1}{5^{p}}+\frac{1}{7^{p}}+\ldots\). to \(\infty\) is convergent for

(a.) \(p>0\)
(b.) \(p>1\)
(c.) \(p>2\)
(d.) \(p<1\)

20. The series \(\frac{1}{1.4}+\frac{1}{2.5}+\frac{1}{3.6}+\ldots\). to \(\infty\) is

(a.) Convergent
(b.) Divergent
(c.) Oscillating
(d.) Converges conditionally