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

1. If \(b_{n}= \begin{cases}\frac{1}{\sqrt{n}} & \text { if } n \text { is odd } \\ \frac{1}{n} & \text { if } n \text { is even }\end{cases}\)

Then

(a.) Both \(\left\langle b_{n}\right\rangle\) and \(\sum b_{n}\) are convergent
(b.) Both \(\left\langle b_{n}\right\rangle\) and \(\sum b_{n}\) are divergent
(c.) \(\left\langle b_{n}\right\rangle\) is convergent but \(\sum b_{n}\) is not
(d.) \(\sum b_{n}\) is convergent but \(\left\langle a_{n}\right\rangle\) is convergent

2. Consider the sequence \(\left\{\frac{(-1)^{n}}{n}\right\}\) and the series \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n}\). Then

(a.) the sequence converges but not the series.
(b.) the series converges but not the sequence.
(c.) neither the series nor the sequence converges.
(d.) both the series and the sequence converge.

3. Which of the following conditions does NOT ensure the convergence of a real sequence \(\left\{a_{n}\right\}\) ?

(a.) \(\left|a_{n}-a_{n+1}\right| \rightarrow 0\) as \(n \rightarrow \infty\)
(b.) \(\sum_{n=1}^{\infty}\left|a_{n}-a_{n+1}\right|\) is convergent
(c.) \(\sum_{n=1}^{\infty} n a_{n}\) is convergent
(d.) The sequence \(\left\{a_{2n}\right\},\left\{a_{2n+1}\right\}\) and \(\left\{a_{3n}\right\}\) are convergent.

4. The series \(\sum\left\{\frac{1}{2^{n-1}}+(-1)^{n-1}\right\}\) is

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

5. The sum of the series \(\frac{1}{1!}+\frac{1+2}{2!}+\frac{1+2+3}{3!}+\ldots\) equals

(a.) \(e\)
(b.) \(\frac{e}{2}\)
(c.) \(\frac{3e}{2}\)
(d.) \(1+\frac{e}{2}\)

6. Let \(\left\{a_{n}\right\}\) be an increasing sequence of positive real numbers such that the series \(\sum_{k=1}^{\infty} a_{k}\) is divergent. Let \(S_{n}=\sum_{k=1}^{n} a_{k}\) for \(n=1,2, \ldots\) and \(t_{n}=\sum_{k=2}^{n} \frac{a_{k}}{S_{k-1} S_{k}}\) for \(n=2,3, \ldots\). Then \(\lim _{n \rightarrow \infty} t_{n}\) is equal to

(a.) \(\frac{1}{a_{1}}\)
(b.) 0
(c.) \(\frac{1}{\left(a_{1}+a_{2}\right)}\)
(d.) \(a_{1}+a_{2}\)

7. In each of the following cases, which of the series is absolutely convergent.

(a.) \(\sum_{n=1}^{\infty}(-1)^{n} \frac{1}{2n+3}\)
(b.) \(\sum_{n=1}^{\infty}(-1)^{n} \frac{n}{n+2}\)
(c.) \(\sum_{n=1}^{\infty}(-1)^{n} \frac{n \log n}{e^{n}}\)
(d.) None of these

8. Let \(\left\{x_{n}\right\}\) be a sequence of real numbers so that \(\sum_{n=1}^{\infty}\left|x_{n}-x\right|=c\), with \(c\) finite. Then

(a.) \(\left\{x_{n}\right\}\) may not be bounded
(b.) \(\left\{x_{n}\right\}\) must converge to \(x\)
(c.) \(\left\{x_{n}\right\}\) must converge to \(x+c\)
(d.) \(\left\{x_{n}\right\}\) is bounded but not necessarily convergent.

9. Let \(\sum_{n=1}^{\infty} a_{n}\) and \(\sum_{n=1}^{\infty} b_{n}\) be two series, where \(a_{n}=\frac{(-1)^{n} n}{2^{n}}\) and \(b_{n}=\frac{(-1)^{n}}{\log (n+1)}\) for all \(n=\mathbb{N}\). Then

(a.) both \(\sum_{n=1}^{\infty} a_{n}\) and \(\sum_{n=1}^{\infty} b_{n}\) are absolutely convergent
(b.) \(\sum_{n=1}^{\infty} a_{n}\) is absolutely convergent but \(\sum_{n=1}^{\infty} b_{n}\) is conditionally convergent
(c.) \(\sum_{n=1}^{\infty} a_{n}\) is conditionally convergent but \(\sum_{n=1}^{\infty} b_{n}\) is absolutely convergent
(d.) both \(\sum_{n=1}^{\infty} a_{n}\) and \(\sum_{n=1}^{\infty} b_{n}\) are conditionally convergent

10. We are given a convergent series \(\sum_{n=1}^{\infty} a_{n}\), where \(a_{n} \geq 0\) for each \(n\). Which of the following correctly describes the behavior of the series \(\sum_{n=1}^{\infty} \frac{\sqrt{a_{n}}}{n^{p}}, \quad 1 \leq p \leq 2\)?

(a.) Diverges when \(p=1\), but converges for \(p \in(1,2]\)
(b.) Converges for every \(p \in[1,2]\)
(c.) Diverges when \(p \in\left[1, \frac{5}{4}\right]\), but converges for \(p \in\left[\frac{5}{4}, 2\right]\)
(d.) Diverges for every \(p \in[1,2]\)