Practice Questions for CSIR NET Real Analysis : Uniform Convergence

1. For \(n \geq 1\), let \(g_n(x)=\sin ^2\left(x+\frac{1}{n}\right), x \in[0, \infty)\) and \(f_n(x)=\int_0^x g_n(t) d t\). Then

(a.) \(\left\{f_n\right\}\) converges pointwise to a function \(f\) on \([0, \infty)\), but does not converge uniformly on \([0, \infty)\).
(b.) \(\left\{f_n\right\}\) does not converge to any function on \([0, \infty)\).
(c.) \(\left\{f_n\right\}\) converges uniformly on \([0,1)\).
(d.) \(\left\{f_n\right\}\) converges uniformly on \([0, \infty)\).

2. Let \(\left\{b_n\right\}\) and \(\left\{c_n\right\}\) be a sequence of real numbers then a necessary and sufficient condition for the sequence of polynomials \(f_n(x)=b_n x+c_n x^2\) to converge uniformly to 0 on the real line is

(a.) \(\lim _{n \rightarrow \infty} b_n=0\) and \(\lim _{n \rightarrow \infty} c_n=0\)
(b.) \(\sum_{n=1}^{\infty}\left|b_n\right|<\infty\) and \(\sum_{n=1}^{\infty}\left|c_n\right|<\infty\)
(c.) There exists a positive integer \(N\) such that \(b_n=0\) and \(c_n=0\) for all \(n>N\)
(d.) \(\lim _{n \rightarrow \infty} c_n=0\)

3. Let \(\left\{f_n\right\}\) be a sequence of continuous functions on \(\mathbb{R}\).

(a.) If \(\left\{f_n\right\}\) converges to \(f\) pointwise on \(\mathbb{R}\) then \(\lim _{n \rightarrow \infty} \int_{-\infty}^{\infty} f_n(x) d x=\int_{-\infty}^{\infty} f(x) d x\)
(b.) If \(\left\{f_n\right\}\) converges to \(f\) uniformly on \(\mathbb{R}\) then \(\lim _{n \rightarrow \infty} \int_{-\infty}^{\infty} f_n(x) d x=\int_{-\infty}^{\infty} f(x) d x\)
(c.) If \(\left\{f_n\right\}\) converges to \(f\) uniformly on \(\mathbb{R}\) then \(f\) is continuous on \(\mathbb{R}\)
(d.) There exists a sequence of continuous functions \(\left\{f_n\right\}\) on \(\mathbb{R}\), such that \(\left\{f_n\right\}\) converges to \(f\) uniformly on \(\mathbb{R}\), but \(\lim _{n \rightarrow \infty} \int_{-\infty}^{\infty} f_n(x) d x \neq \int_{-\infty}^{\infty} f(x) d x\)

4. Let \(P_n(x)=a_n x^2+b_n x+c_n\) be a sequence of quadratic polynomials where \(a_n, b_n, c_n \in \mathbb{R}\) for all \(n \geq 1\). Let \(\lambda_0, \lambda_1, \lambda_2\) be distinct real numbers such that \(\lim _{n \rightarrow \infty} P_n\left(\lambda_0\right)=A_0\), \(\lim _{n \rightarrow \infty} P_n\left(\lambda_1\right)=A_1\), and \(\lim _{n \rightarrow \infty} P_n\left(\lambda_2\right)=A_2\). Then

(a.) \(\lim _{n \rightarrow \infty} P_n(x)\) exists for all \(x \in \mathbb{R}\)
(b.) \(\lim _{n \rightarrow \infty} P_n^\prime(x)\) exists for all \(x \in \mathbb{R}\)
(c.) \(\lim _{n \rightarrow \infty} P_n\left(\frac{\lambda_0+\lambda_1+\lambda_2}{3}\right)\) does not exist
(d.) \(\lim _{n \rightarrow \infty} P_n^\prime\left(\frac{\lambda_0+\lambda_1+\lambda_2}{3}\right)\) does not exist

5. Let \(f_n:[1,2] \rightarrow[0,1]\) be given by \(f_n(x)=(2-x)^n\) for all non-negative integers \(n\). Let \(f(x)=\lim _{n \rightarrow \infty} f_n(x)\) for \(1 \leq x \leq 2\). Then which of the following is true?

(a.) \(f\) is a continuous function on \([1,2]\)
(b.) \(f_n\) converges uniformly to \(f\) on \([1,2]\) as \(n \rightarrow \infty\)
(c.) \(\lim _{n \rightarrow \infty} \int_1^2 f_n(x) d x=\int_1^2 f(x) d x\)
(d.) for any \(a \in(1,2)\) we have \(\lim _{n \rightarrow \infty} f_n^\prime(a) \neq f^\prime(a)\)

6. Let \(f_n(x)=\left\{\begin{array}{cc}1-n x & \text { for } x \in\left[0, \frac{1}{n}\right] \\ 0 & \text { for } x \in\left[\frac{1}{n}, 1\right]\end{array}\right.\) Then

(a.) \(\lim _{n \rightarrow \infty} f_n(x)\) defines a continuous function on \([0,1]\)
(b.) \(\left\{f_n\right\}\) converges uniformly on \([0,1]\)
(c.) \(\lim _{n \rightarrow \infty} f_n(x)=0\) for all \(x \in[0,1]\)
(d.) \(\lim _{n \rightarrow \infty} f_n(x)\) exists for all \(x \in[0,1]\)

7. Let \(f_n(x)=x^{1/n}\) for \(n \in[0,1]\). Then

(a.) \(\lim _{n \rightarrow \infty} f_n(x)\) exists for all \(x \in[0,1]\)
(b.) \(\lim _{n \rightarrow \infty} f_n(x)\) defines a continuous function on \([0,1]\)
(c.) \(\left\{f_n\right\}\) converges uniformly on \([0,1]\)
(d.) \(\lim _{n \rightarrow \infty} f_n(x)=0\) for all \(x \in[0,1]\)

8. Let \(f(x)=\sum_{n=1}^{\infty} \frac{\sin (n x)}{n^2}\). Then

(a.) \(\lim _{x \rightarrow 0} f(x)=0\)
(b.) \(\lim _{x \rightarrow 0} f(x)=1\)
(c.) \(\lim _{x \rightarrow 0} f(x)=\frac{\pi^2}{6}\)
(d.) \(\lim _{x \rightarrow 0} f(x)\) does not exist

9. The series \(\sum_{m=1}^{\infty} x^{\ln m}\), \(x>0\), is convergent on the interval

(a.) \((0,1 / e)\)
(b.) \((1 / e, e)\)
(c.) \((0, e)\)
(d.) \((1, e)\)

10. Let \(f_n(x)=\frac{x}{\{(n-1) x+1\}\{n x+1\}}\) and

(a.) Converges uniformly on \([0,1]\)
(b.) Converges pointwise on \([0,1]\) but not uniformly
(c.) Converges pointwise for \(x=0\) but not for \(x \in (0,1]\)
(d.) Does not converge for \(x \in [0,1]\)

11. Let \(\left\{f_n\right\}\) be a sequence of real-valued differentiable function on \([a, b]\) such that \((-1,1)\) \(f_n(x) \rightarrow f(x)\) as \(n \rightarrow \infty\) for every \(x \in [a, b]\) and for some Riemann-integrable function \(f:[a, b] \rightarrow \mathbb{R}\). Consider the statements:

\(P_1:\left\{f_n\right\}\) converges uniformly

\(P_2:\left\{f_n^{\prime}\right\}\) converges uniformly

\(P_3: \int_a^b f_n(x) d x \rightarrow \int_a^b f(x) d x\)

\(P_4: f\) is differentiable

Then which one of the following need NOT be true

(a.) \(P_1\) implies \(P_3\)
(b.) \(P_2\) implies \(P_1\)
(c.) \(P_2\) implies \(P_4\)
(d.) \(P_3\) implies \(P_1\)

12. Let \(f_n(x)=\frac{x^n}{1+x}\) and \(g_n(x)=\frac{x^n}{1+n x}\) for \(x \in \mathbb{N}\). Then on the interval \([0,1]\),

(a.) both \(\left\{f_n\right\}\) and \(\left\{g_n\right\}\) converge uniformly
(b.) neither \(\left\{f_n\right\}\) nor \(\left\{g_n\right\}\) converges uniformly
(c.) \(\left\{f_n\right\}\) converges uniformly but \(\left\{g_n\right\}\) does not converge uniformly
(d.) \(\left\{g_n\right\}\) converges uniformly but \(\left\{f_n\right\}\) does not converge uniformly

13. Which of the following sequences \(\left\{f_n\right\}_{n=1}^{\infty}\) of functions does NOT converge uniformly on \([0,1]\)?

(a.) \(f_n(x)=\frac{e^{-x}}{n}\)
(b.) \(f_n(x)=(1-x)^n\)
(c.) \(f_n(x)=\frac{x^2+n x}{n}\)
(d.) \(f_n(x)=\frac{\sin (n x+n)}{n}\)

14. Which one of the following statements holds?

(a.) The series \(\sum_{n=0}^{\infty} x^n\) converges for each for \(x \in [-1,1]\)
(b.) The series \(\sum_{n=0}^{\infty} x^n\) converges uniformly in \(x \in (-1,1)\)
(c.) The series \(\sum_{n=1}^{\infty} \frac{x^n}{n^2}\) converges for each \(x \in [-1,1]\)
(d.) The series \(\sum_{n=1}^{\infty} \frac{x^n}{n^2}\) converges uniformly in \((-1,1)\)

15. Consider two sequences \(\left\{f_n\right\}\) and \(\left\{g_n\right\}\) of functions where \(f_n:[0,1] \rightarrow \mathbb{R}\) and \(g_n: \mathbb{R} \rightarrow \mathbb{R}\) are defined by \(f_n(x)=x^n\) and \(g_n(x)=\left\{\begin{array}{ccc}\cos \left(\frac{x-n \pi}{2}\right) & \text{If} & x \in [n-1, n+1] \\ 0 & \text{otherwise}\end{array}\right.\) Then

(a.) Neither \(\left\{f_n\right\}\) nor \(\left\{g_n\right\}\) is uniformly convergent
(b.) \(\left\{f_n\right\}\) is not uniformly convergent but \(\left\{g_n\right\}\) is
(c.) \(\left\{g_n\right\}\) is not uniformly convergent but \(\left\{f_n\right\}\) is
(d.) Both \(\left\{f_n\right\}\) and \(\left\{g_n\right\}\) are uniformly convergent

16. Let \(f_n(x)=\frac{\sin x}{\sqrt{n}}\), \(n=1,2, \ldots\), and \(x \in [-1,1]\). Choose incorrect

(a.) \(\left\{f_n(x)\right\}_{n=1}^{\infty}\) does not converge uniformly in \([-1,1]\)
(b.) \(\lim _{n \rightarrow \infty} \int_{-1}^1 f_n(x) d x \neq 0\)
(c.) \(\left\{f_n^{\prime}(x)\right\}\) does not converge uniformly in \([-1,1]\)
(d.) \(f_x(x)\), \(n=1,2\), is not uniformly continuous in \([-1,1]\)

17. Find out which of the following series converge uniformly for \(x \in (-\pi, \pi)\).

(a.) \(\sum_{n=1}^{\infty} \frac{e^{-n|x|}}{n^3}\)
(b.) \(\sum_{n=1}^{\infty} \frac{\sin (x n)}{n^5}\)
(c.) \(\sum_{n=1}^{\infty}\left(\frac{x}{n}\right)^n\)
(d.) \(\sum_{n=1}^{\infty} \frac{1}{((x+\pi) n)^2}\)

18. Let \(f_n(x)=(-x)^n\), \(x \in [0,1]\). Then decide which of the following are true.

(a.) There exists a pointwise convergent subsequence of \(f_n\)
(b.) \(f_n\) has non-pointwise convergent subsequence.
(c.) \(f_n\) converges pointwise everywhere.
(d.) \(f_n\) has exactly one pointwise convergent subsequence.

19. Consider all sequences \(\left\{f_n\right\}\) of real-valued continuous functions on \([0, \infty)\). Identify which of the following statements are correct.

(a.) If \(\left\{f_n\right\}\) converges to \(f\) pointwise on \([0, \infty)\) then \(\lim _{n \rightarrow \infty} \int_0^{\infty} f_n(x) d x=\int_0^{\infty} f(x) d x\)
(b.) If \(\left\{f_n\right\}\) converges to \(f\) uniformly on \([0, \infty)\) then \(\lim _{n \rightarrow \infty} \int_0^{\infty} f_n(x) d x=\int_0^{\infty} f(x) d x\)
(c.) If \(\left\{f_n\right\}\) converges to \(f\) uniformly on \([0, \infty)\) then \(f\) is continuous on \([0, \infty)\).
(d.) There exists a sequence of continuous functions \(\left\{f_n\right\}\) on \([0, \infty)\) such that \(\left\{f_n\right\}\) converges to \(f\) uniformly on \([0, \infty)\) but \(\lim _{n \rightarrow \infty} \int_0^{\infty} f_n(x) d x \neq \int_0^{\infty} f(x) d x\)

20. Which one of the following statements is true for the sequence of functions \(f_n(x)=\frac{1}{n^2+x^2}\), \(n=1,2, \cdots\), \(x \in [1/2,1]\)?

(a.) The sequence is monotonic and has 0 as the limit for all \(x \in [1/2,1]\) as \(n \rightarrow \infty\)
(b.) The sequence is not monotonic but has \(f(x)=\frac{1}{x^2}\) as the limit as \(n \rightarrow \infty\)
(c.) The sequence is monotonic and has \(f(x)=\frac{1}{x^2}\) as the limit as \(n \rightarrow \infty\)
(d.) The sequence is not monotonic but has 0 as the limit

21. Consider the power series \(f(x)=\sum_{n=2}^{\infty} \log (n) x^n\). The radius of convergence of the series \(f(x)\) is

(a.) 0
(b.) 1
(c.) 3
(d.) \(\infty\)

22. For \(n \geq 1\), let \(f_n(x)=x e^{-n x^2}\), \(x \in \mathbb{R}\). Then the sequence \(\left\{f_n\right\}\) is

(a.) Uniformly convergent on \(\mathbb{R}\)
(b.) Uniformly convergent only on compact subsets of \(\mathbb{R}\)
(c.) Bounded and not uniformly convergent on \(\mathbb{R}\)
(d.) A sequence of unbounded functions

23. Let \(f_n(x)=\frac{1}{1+n^2 x^2}\) for \(n \in \mathbb{N}\), \(x \in \mathbb{R}\). Which of the following are true?

(a.) \(f_n\) converges pointwise on \([0,1]\) to a continuous function
(b.) \(f_n\) converges uniformly on \([0,1]\)
(c.) \(f_n\) converges uniformly on \(\left[\frac{1}{2}, 1\right]\)
(d.) \(\lim _{n \rightarrow \infty} \int_0^1 f_n(x) d x=\int_0^1 \left(\lim _{n \rightarrow \infty} f_n(x)\right) d x\)