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

31. Choose the correct answer, where \(a_{n} \geq 0\)

(a.) If \(\sum a_{n}\) is convergent, then \(\sum a_{n}^{2}\) is also convergent.
(b.) If \(\sum a_{n}\) is convergent, then \(\sum \sqrt{a_{n}}\) is also convergent.
(c.) If \(\sum a_{n}\) is convergent, then \(\sum \frac{a_{n}}{1+a_{n}}\) is also convergent.
(d.) If \(\sum a_{n}\) is convergent (provided \(a_{n} \neq 1\)), then \(\sum \frac{a_{n}}{1-a_{n}}\) is also convergent.

32. Let \(\sum_{n=1}^{\infty} x_n\) be a series of real numbers. Which of the following is true?

(a.) If \(\sum_1^{\infty} x_n\) is divergent, then \(\left\langle x_n\right\rangle\) does not converge to 0.
(b.) If \(\sum_{n=1}^{\infty} x_n\) is convergent, then \(\sum_{n=1}^{\infty} x_n\) is absolutely convergent.
(c.) If \(\sum_{n=1}^{\infty} x_n\) is convergent, then \(x_n^2 \rightarrow 0\) as \(n \rightarrow \infty\).
(d.) If \(x_n \rightarrow 0\), then \(\sum_{n=1}^{\infty} x_n\) is convergent.

33. If \(\sum a_n\) be a convergent series and \(a_n>0\), \(\forall n \in \mathbb{N}\), then the correct statement is:

(a.) \(\sum \sqrt{a_n a_{n+1}}\) is convergent.
(b.) \(\sum a_n^2\) is convergent.
(c.) \(\sum \frac{\sqrt{a_n}}{n}\) is convergent.
(d.) None of the above.

34. Let \(\left\langle a_n\right\rangle\) and \(\left\langle b_n\right\rangle\) be two sequences of real numbers such that \(a_n=b_n-b_{n+1}\) for \(n \in \mathbb{N}\). Then

(a.) Convergence of \(\sum a_n\) implies the convergence of \(\left\langle b_n\right\rangle\), not conversely.
(b.) Convergence of \(\sum a_n\) implies the convergence of \(\sum b_n\), but not conversely.
(c.) Convergence of \(\sum a_n\) implies and is implied by the convergence of \(\left\langle b_n\right\rangle\).
(d.) Convergence of \(\sum b_n\) implies and is implied by the convergence of \(\left\langle a_n\right\rangle\).

35. The alternating series \(\sum_{n=1}^{\infty}(-1)^{n-1} u_n\) converges if \(\left\{u_n\right\}\) is

(a.) A decreasing sequence that converges to zero.
(b.) An increasing sequence that converges to zero.
(c.) A monotone sequence that converges to zero.
(d.) A strictly monotone sequence that converges to zero.

36. Which one of the following series is convergent?

(a.) \(\frac{1}{2 \sqrt{2}}+\frac{1}{3 \sqrt{3}}+\frac{1}{4 \sqrt{4}}+\ldots\)
(b.) \(1 \frac{1}{2}-1 \frac{1}{3}+1 \frac{1}{4}-1 \frac{1}{5}+\ldots\)
(c.) \(\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\ldots\)
(d.) \(x+x^2+x^3+x^4+\ldots\) for \(|x|<1\)

37. Select the incorrect statements

(a.) A finite set has an infinite number of limit points.
(b.) The sequence \(1+r+r^2+\ldots+r^n\), where \(-1 \leq r<1\) and \(n \in \mathbb{N}\), is bounded above but not bounded below.
(c.) If \(u_n>0\) for all \(n\), then \(\sum u_n\) and \(\sum \frac{1}{u_n}\) converge or diverge together.
(d.) The set of real numbers except the integers which are multiples of 2 is an uncountable set.

38. The set of all positive values of \(a\) for which the series \(\sum_{n=1}^{\infty}\left(\frac{1}{n}-\tan ^{-1}\left(\frac{1}{n}\right)\right)^a\) converges, is

(a.) \(\left(0, \frac{1}{3}\right]\)
(b.) \(\left(0, \frac{1}{3}\right)\)
(c.) \(\left[\frac{1}{3}, \infty\right)\)
(d.) \(\left(\frac{1}{3}, \infty\right)\)

39. Let \(S\) be the series \(\sum_{k=1}^{\infty} \frac{1}{(2 k-1) 2^{(2 k-1)}}\) and \(T\) be the series \(\sum_{k=3}^{\infty}\left(\frac{3 k-4}{3 k+2}\right)^{\frac{(k+1)}{3}}\) of real numbers. Then which one of the following is TRUE?

(a.) Both the series \(S\) and \(T\) are convergent.
(b.) \(S\) is convergent and \(T\) is divergent.
(c.) \(S\) is divergent and \(T\) is convergent.
(d.) Both the series \(S\) and \(T\) are divergent.

40. For \(n \geq 1\), let

\(a_{n}=\begin{cases}n 2^{-n}, & \text{if \(n\) is odd}, \\ -3^{-n}, & \text{if \(n\) is even}\end{cases}\)

Which of the following statements is (are) TRUE?

(a.) The sequence \(\left\{a_{n}\right\}\) converges.
(b.) The sequence \(\left\{\left|a_{n}\right|^{1 / n}\right\}\) converges.
(c.) The series \(\sum_{n=1}^{\infty} a_{n}\) converges.
(d.) The series \(\sum_{n=1}^{\infty}\left|a_{n}\right|\) converges.

41. Let

\(A=\{n \in \mathbb{N}: n=1\) or the only prime factors of \(n\) are 2 or 3\(\}\)

For example, \(6 \in A\) and \(10 \notin A\)

Let \(S=\sum_{n \in A} \frac{1}{n}\). Then

(a.) \(A\) is finite.
(b.) \(S\) is a divergent series.
(c.) \(S=3\).
(d.) \(S=6\).

42. Which of the following convergent?

(a.) \(\sum_{n=1}^{\infty} n^{2} n^{-n}\)
(b.) \(\sum_{n=1}^{\infty} n^{-2} 2^{n}\)
(c.) \(\sum_{n=1}^{\infty} \frac{1}{n \log n}\)
(d.) \(\sum_{n=1}^{\infty} \frac{1}{n \log (1+1 / n)}\)

43. Evaluate \(\lim _{n \rightarrow \infty} \sum_{k=0}^{n} \frac{n}{k^{2}+n^{2}}\).

(a.) \(\frac{\pi}{2}\)
(b.) \(\pi\)
(c.) \(\frac{\pi}{8}\)
(d.) \(\frac{\pi}{4}\)