Function Sequences and Uniform Convergence - Real Analysis - Mathematics Previous Year Questions

1

2008 · Mathematics · Real Analysis · Function Sequences and Uniform Convergence

Mathematics (MA) 2008

Which one of the following statements holds?

A

The series \(\sum_{n=0}^{\infty} x^n\) converges for each \(x \in [-1, 1]\)

B

The series \(\sum_{n=1}^{\infty} x^n\) converges uniformly in \((-1, 1)\)

C

The series \(\sum_{n=1}^{\infty} \frac{x^n}{n}\) converges for each \(x \in [-1, 1]\)

D

The series \(\sum_{n=1}^{\infty} \frac{x^n}{n^2}\) converges uniformly in \((-1, 1)\)

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2

2008 · Mathematics · Real Analysis · Function Sequences and Uniform Convergence

Mathematics (MA) 2008

For \(x \in [-\pi, \pi]\), let \(f(x) = (\pi + x)(\pi - x)\) and \(g(x) = \begin{cases} \cos(1/x) & \text{if } x \neq 0, \\ 0 & \text{if } x = 0. \end{cases}\) Consider the statements \(P\): The Fourier series of \(f\) converges uniformly to \(f\) on \([-\pi, \pi]\). \(Q\): The Fourier series of \(g\) converges uniformly to \(g\) on \([-\pi, \pi]\). Then

A

\(P\) and \(Q\) are true

B

\(P\) is true but \(Q\) is false

C

\(P\) is false but \(Q\) is true

D

both \(P\) and \(Q\) are false

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3

2008 · Mathematics · Real Analysis · Function Sequences and Uniform Convergence

Mathematics (MA) 2008

Which one of the following is true?

A

\(C^{\infty}([0, 1])\) is dense in \(X\)

B

\(X\) is dense in \(L^{\infty}([0, 1])\)

C

\(X\) has a countable basis

D

There is a sequence in \(X\) which is uniformly Cauchy on \([0, 1]\) but does not converge uniformly on \([0, 1]\)

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4

2008 · Mathematics · Real Analysis · Function Sequences and Uniform Convergence

Mathematics (MA) 2008

Then

A

\(F\) and \(f\) are continuous

B

\(F\) is continuous and \(f\) is discontinuous

C

\(F\) is discontinuous and \(f\) is continuous

D

\(F\) and \(f\) are discontinuous

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5

2009 · Mathematics · Real Analysis · Function Sequences and Uniform Convergence

Mathematics (MA) 2009

Which of the following sequence \(\{f_n\}_{n=1}^{\infty}\) 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

\(f_n(x) = \frac{x^2 + nx}{n}\)

D

\(f_n(x) = \frac{\sin(nx + n)}{n}\)

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6

2009 · Mathematics · Real Analysis · Function Sequences and Uniform Convergence

Mathematics (MA) 2009

Let \( f_n(x) = \frac{1}{n} \sum_{k=1}^n \sqrt{k(n-k)} \binom{n}{k} x^k (1-x)^{n-k} \) for \( x \in [0,1], \ n = 1, 2, \ldots \). If \( \lim_{n \to \infty} f_n(x) = f(x) \) for \( x \in [0,1] \), then the maximum value of \( f(x) \) on \( [0,1] \) is

A

1

B

\(\frac{1}{2} \)

C

\(\frac{1}{3} \)

D

\(\frac{1}{4} \)

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7

2010 · Mathematics · Real Analysis · Function Sequences and Uniform Convergence

Mathematics (MA) 2010

Let \( \{ f_n \} \) be a sequence of real valued differentiable functions on \( [a,b] \) such that \( f_n(x) \to f(x) \) as \( n \to \infty \) for every \( x \in [a,b] \) and for some Riemann-integrable function \( f : [a,b] \to \mathbb{R} \). Consider the statements
\( P_1 : \{ f_n \} \) converges uniformly
\( P_2 : \{ f_n' \} \) converges uniformly
\( P_3 : \int_a^b f_n(x) dx \to \int_a^b f(x) dx \)
\( P_4 : f \) is differentiable
Then which one of the following need NOT be true

A

\(P_1 \) implies \(P_3 \)

B

\(P_2 \) implies \(P_3 \)

C

\(P_2 \) implies \(P_4 \)

D

\(P_3 \) implies \(P_4 \)

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8

2010 · Mathematics · Real Analysis · Function Sequences and Uniform Convergence

Mathematics (MA) 2010

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

A

both \(\{ f_n \} \) and \(\{ g_n \} \) converge uniformly

B

neither \(\{ f_n \} \) nor \(\{ g_n \} \) converges uniformly

C

\(\{ f_n \} \) converges uniformly but \(\{ g_n \} \) does not converge uniformly

D

\(\{ g_n \} \) converges uniformly but \(\{ f_n \} \) does not converge uniformly

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9

2010 · Mathematics · Real Analysis · Function Sequences and Uniform Convergence

Mathematics (MA) 2010

Consider the power series \( \sum_{n=1}^\infty \frac{x^n}{\sqrt{n}} \) and \( \sum_{n=1}^\infty \frac{x^n}{n} \). Then

A

both converge on \([-1,1] \)

B

both converge on \([-1,1) \)

C

exactly one of them converges on \((-1,1) \)

D

none of them converges on \([-1,1) \)

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10

2010 · Mathematics · Real Analysis · Function Sequences and Uniform Convergence

Mathematics (MA) 2010

The values of \( a_0 \) and \( b_0 \) respectively are

A

\(\frac{1}{2} \) and \(\frac{1}{2} \)

B

\(\frac{1}{12} \) and \(-\frac{1}{12} \)

C

\(\frac{1}{2} \) and \(\frac{1}{12} \)

D

\(\frac{1}{2} \) and \(-\frac{1}{12} \)

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11

2011 · Mathematics · Real Analysis · Function Sequences and Uniform Convergence

Mathematics (MA) 2011

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

A

\((0, 1/e)\)

B

\((1/e, e)\)

C

\((0, e)\)

D

\((1, e)\)

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12

2011 · Mathematics · Real Analysis · Function Sequences and Uniform Convergence

Mathematics (MA) 2011

For \(n \geq 1\), let \(\{X_n\}\) be a sequence of independent random variables with \[P(X_n = n) = P(X_n = -n) = \frac{1}{2n^2}, \quad P(X_n = 0) = 1 - \frac{1}{n^2}.\] Then, which of the following statements is TRUE for the sequence \(\{X_n\}\)?

A

Weak Law of Large Numbers holds but Strong Law of Large Numbers does not hold

B

Weak Law of Large Numbers does not hold but Strong Law of Large Numbers holds

C

Both Weak Law of Large Numbers and Strong Law of Large Numbers hold

D

Both Weak Law of Large Numbers and Strong Law of Large Numbers do not hold

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13

2011 · Mathematics · Real Analysis · Function Sequences and Uniform Convergence

Mathematics (MA) 2011

The sequence \( \{s_n\} \)

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 \(x \in [0, 1] \)

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14

2013 · Mathematics · Real Analysis · Function Sequences and Uniform Convergence

Mathematics (MA) 2013

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

A

\(\lim_{x \to 0} f(x) = 0\)

B

\(\lim_{x \to 0} f(x) = 1\)

C

\(\lim_{x \to 0} f(x) = \pi^2/6\)

D

\(\lim_{x \to 0} f(x)\) does not exist

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15

2013 · Mathematics · Real Analysis · Function Sequences and Uniform Convergence

Mathematics (MA) 2013

Let \(x_0 = 0\). Define \(x_{n+1} = \cos x_n\) for every \(n \ge 0\). Then

A

\(\{x_n\}\) is increasing and convergent

B

\(\{x_n\}\) is decreasing and convergent

C

\(\{x_n\}\) is convergent and \(x_{2n} < \lim_{m \to \infty} x_m < x_{2n+1}\) for every \(n \in \mathbb{N}\)

D

\(\{x_n\}\) is not convergent

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16

2013 · Mathematics · Real Analysis · Function Sequences and Uniform Convergence

Mathematics (MA) 2013

Let \(\{a_n\}\) be the sequence of consecutive positive solutions of the equation \(\tan x = x\) and let \(\{b_n\}\) be the sequence of consecutive positive solutions of the equation \(\tan \sqrt{x} = x\). Then

A

\(\sum_{n=1}^{\infty} \frac{1}{a_n}\) converges but \(\sum_{n=1}^{\infty} \frac{1}{b_n}\) diverges

B

\(\sum_{n=1}^{\infty} \frac{1}{a_n}\) diverges but \(\sum_{n=1}^{\infty} \frac{1}{b_n}\) converges

C

Both \(\sum_{n=1}^{\infty} \frac{1}{a_n}\) and \(\sum_{n=1}^{\infty} \frac{1}{b_n}\) converge

D

Both \(\sum_{n=1}^{\infty} \frac{1}{a_n}\) and \(\sum_{n=1}^{\infty} \frac{1}{b_n}\) diverge

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17

2013 · Mathematics · Real Analysis · Function Sequences and Uniform Convergence

Mathematics (MA) 2013

The value of the limit \(\lim_{n \to \infty} \frac{2^{-n^2}}{\sum_{k=n+1}^{\infty} 2^{-k^2}}\) is

A

0

B

some \(c \in (0, 1)\)

C

1

D

\(\infty\)

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18

2013 · Mathematics · Real Analysis · Function Sequences and Uniform Convergence

Mathematics (MA) 2013

Let \(S = \{x \in \mathbb{R} : x \ge 0, \sum_{n=1}^{\infty} x^{\sqrt{n}} < \infty\}\). Then the supremum of \(S\) is

A

1

B

\(\frac{1}{e}\)

C

0

D

\(\infty\)

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19

2013 · Mathematics · Real Analysis · Function Sequences and Uniform Convergence

Mathematics (MA) 2013

The value of \(n_0\) is ______

Check answer

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20

2014 · Mathematics · Real Analysis · Function Sequences and Uniform Convergence

Mathematics (MA) 2014

If \(X_1, X_2\) is a random sample of size 2 from an \(N(0,1)\) population, then \(\frac{(X_1 + X_2)^2}{(X_1 - X_2)^2}\) follows

A

\(\chi^2_{(2)}\)

B

\(F_{2,2}\)

C

\(F_{2,1}\)

D

\(F_{1,1}\)

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