RGU Question Paper Mathematics Semester IV 2020: Real Analysis-I

 

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BSc (IV Semester) Examination

MATHEMATICS

Paper : MATH–241 (New Course)

( Real Analysis—I )

Full Marks : 80

Pass Marks : 35%

Time : Three Hours

Note : 1. Answer all questions.
2. The figures in the margin indicate full marks for the questions.

1. Answer any five parts : 2×5=10

(a) Give an example of set which is—
(i) bounded below but not bounded above;
(ii) bounded above but not bounded below. 1+1=2

(b) Define a monotonic sequence. Can a monotonic sequence oscillate?

(c) Define an open set. Let
$S_n=\left\{\left(-\frac{1}{n},\frac{1}{n}\right): n\in \mathbb{N}\right\}$
Is $\cap S_n$ an open set?

(d) Show that the function
$f\left(x\right)=\left\{\begin{array}{ll}\frac{\sin 2x}{x}& when x\ne 0\\ 1& when x=0\end{array}\right.$
is discontinuous at x = 0. What kind of discontinuity $f$ has at $x=0$?

(e) Show that the function $f$ defined on $\mathbb{R}$ by
$f\left(x\right)=\left\{\begin{array}{ll}x& when 0\le x<1\\ 1& when x\ge 1\end{array}\right.$
is not differentiable at $x=1$.

(f) Show that
$\underset{n\to \infty }{\lim }\left[\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+...+\frac{1}{\sqrt{n^2+n}}\right]=1$

(g) Investigate the convergence of the series
$\sum _{n=1}^{\infty }\sin \frac{1}{n}$

(h) Find the infimum and supremum of the set
$\left\{\frac{(-1{)}^n}{n}:n\in \mathbb{N}\right\}$

(i) State D’Alembert’s ratio test.

2. Answer any four parts : 5×4=20

(a) Show that the series $\left\{S_n\right\}$ defined by the recursive formula
$S_{n+1}=\sqrt{3S_n}, S_1=1$
converges to $3$.

(b) If a function $f$ is continuous on $[a,b]$and $f\left(a\right)\ne f\left(b\right)$, then prove that $f$
assumes every value between $f(a)$and $f\left(b\right)$.

(c) Evaluate
$\underset{x\to 0}{\lim }\left(\frac{x-tanx}{x^3}\right)$

(d) Test the convergence of the series
$\frac{1·2}{3^2·4^2}+\frac{3·4}{5^2·6^2}+\frac{5·6}{7^2·8^2}+...$

(e) Prove that the set of real numbers $\mathbb{R }$is uncountable.

(f) Define a Cauchy sequence. Show that every convergent sequence of real numbers is a Cauchy sequence.

(g) Define a bounded sequence. If $\{a_n\}$and $\left\{b_n\right\}$ are two bounded sequences, prove that
$\underset{\_}{lim}{a}_n+\overline{lim}{b}_n\le \overline{lim}(a_n+b_n)$

(h) Examine the validity of hypothesis and conclusion of the Rolle’s theorem for the function
$f\left(x\right)=2x^3+x^2-4x-2$
on $[-2,2]$.

3. Answer any five parts : 10×5=50

(a) (i) Test the convergence of the series
$\sum _{n=1}^{\infty }\frac{n^2-1}{n^2+1}{x}^n$, 5

(ii) Show that the sequence $\{S_n\}$given by
$S_n=\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{n!}, \forall n\in \mathbb{N}$
is convergent. 5

(b) (i) Prove that every absolutely convergent series is convergent. Is converse true? Justify. 5

(ii) Let $f\left(x\right)=x^2, \forall x\in \mathbb{R}$. Show that $f$ is uniformly continuous on $[0,1]$. 5

(c) Prove that a bounded infinite set of real numbers has at least one limit point. 10
(d) (i) If
$f(x+h)=f\left(x\right)+hf\text{'}\left(x\right)+\frac{h^2}{2!}f\text{'}\text{'}(x+\theta h)$
then find the value of $\theta$ as $x\to a$, where $f\left(x\right)=(x-a{)}^{5/2}$. 5

(ii) State Langrange’s mean value theorem. If a function $f(x)$satisfies the conditions of the Langrange’s mean value theorem and $f\text{'}\left(x\right)=0$ for all $x\in [a,b]$, then prove that $f\left(x\right)$ is constant on $[a,b]$. 5

(e) (i) Define a uniformly continuous function. Show that a function which is uniformly continuous on an interval is continuous on that interval. 5

(ii) For any fixed value of $x$, show that the series
$\sum _{n=1}^{\infty }\frac{\sin nx}{n^2}$
converges. 5

(f) (i) Examine the validity of hypothesis and conclusion of the Lagrange’s mean value theorem for the function $f\left(x\right)=x(x-1)(x-2)$on $[0,\frac{1}{2}]$.  5
(ii) If a function $f$ is continuous on $[a,b]$, derivable at $(a,b)$ and $f\text{'}\left(x\right)>0, \forall x\in (a,b)$, then show that $f$ is a strictly increasing function. 5
(g) Show that the series
$\sum _{n=1}^{\infty }\frac{3·6·9···3n}{7·10·13···(3n+4)}{x}^n, x>0$,
converges for $x\le 1$ and diverges for $x>1$. 10

(h) (i) Show that the sequence $\{a_n\}$defined by
$a_{n+1}=\frac{1}{2}\left(a_n+\frac{9}{a_n}\right), n\ge 1 and a_i>0$
converges to $3$. 5

(ii) Expand $\sin x$ in powers of $(x-\frac{\pi }{2})$by Taylor’s theorem. 5

(i) State Raabe’s logarithmic and Gauss’s test for convergence. Test the convergence of the series
$\sum _{n=1}^{\infty }\frac{\sqrt{n}}{1+n^3}$. 10