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IGNOU BMTC 133 Real Analysis Solved Assignment 2024

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Last Date of Submission of IGNOU BMTC-133 (BAG) 2024 Assignment is for January 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).
Semester Wise
January 2024 Session: 30th March, 2024 (for June 2024 Term End Exam).
July 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).

Title Name	ignou BMTC 133 solved assignment 2024
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BAG
Course Name	BACHELOR OF ARTS
Subject Code	BMTC 133
Subject Name	Real Analysis
Year	2024
Session	-
Language	English Medium
Assignment Code	BMTC-133/Assignmentt-1//2024
Product Description	Assignment of BAG (BACHELOR OF ARTS) 2024. Latest BMTC 133 2024 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU BMTC-133 (BAG) 2024 Assignment is for January 2024 Session: 30th September, 2024 (for December 2024 Term End Exam). Semester Wise January 2024 Session: 30th March, 2024 (for June 2024 Term End Exam). July 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).
Assignment Code	BMTC 133/2024

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Questions Included in this Help Book

Ques 1.

Which of the following statements are true or false? Give reasons for your answers in the form of a short proof or counter-example, whichever is appropriate:

i) Every infinite set is an open set.

ii) The negation of p∧ ~ q is p → q.

iii) −1is a limit point of the interval ]−2 ,1],

iv) The necessary condition for a function $f$ to be integrable is that it is continuous.

v) The function $f:\mathbb{R}\rightarrow \mathbb{R}$ defined by $f(x)=\left | x-2 \right |+\left | 3-x \right |$ is differentiable at $x=5.$

Ques 2.

Give an example for each of the following.

i) A set in $\mathbb{R}$ with a unique limit point.

ii) A set in $\mathbb{R}$ whose all points except the one are its limit points.

iii) A set having no limit point.

iv) A set S with $S^{\circ}=\bar{S}.$

v) A bijection from $\mathbb{N}_{odd}$ to $\mathbb{Z}.$

Ques 3.

Give an example of a divergent sequence which has two convergent subsequences. Justify your claim.

Ques 4.

The product of two divergent sequences is divergent. True or false? Justify.

Ques 5.

Let $(a_{n})_{n\in \, \mathbb{N}}$ be any sequence. Show that $\lim_{n\rightarrow \infty }a_{n}=L$ iff for every ε > 0, there exists some $N\in \! \mathbb{N}$ such that n ≥ N implies $a_{n}\in N_{\varepsilon }(L).$

Ques 6.

d) Show that $\left ( \frac{1}{n^{2}+n+1} \right )_{n\in \mathbb{N}}$ is a Cauchy sequence.

Ques 7.

Evaluate

$\lim_{n\rightarrow \infty }$ $\,$ $\left [ \frac{n}{1+n^{2}}+\frac{n}{4+n^{2}}+\frac{n}{9+n^{2}}+\cdots+\frac{n}{2n^{2}} \right ].$

Ques 8.

Determine the points of discontinuity of the function f and the nature of discontinuity at each of those points:

$\left\{\begin{matrix} -x^{2}\: , &when\: x\leq 0 \\4-5x, &when\: 0< x\leq 1 \\3x-4x^{2}\: , &when\: 1< x\leq 2 \\ -12x+2x\: , & when\: x< 2 \end{matrix}\right.$

Also check whether the function f is derivable at x = .1

Ques 9.

Find the following limit

$\lim_{x\rightarrow 0}\frac{1-cos\: x^{2}}{x^{2}sin\: x^{2}}$

Ques 10.

Prove that a strictly decreasing function is always one-one

Ques 11.

Determine the local minimum and local maximum values of the function f defined by $f(x)=3-5x^{3}+5x^{4}-x^{5}\: .$

Ques 12.

Let $f:[01]\rightarrow$ $\mathbb{R}$ be a function defined by $f(x)=x^{m}(1-x)^{n}\: ,$ where $m,n\in \mathbb{N}.$ Find the values of m and n such that the Rolle’s Theorem holds for the function $f.$

Ques 13.

Let $f$ be a differentiable function on $[\alpha ,\beta ]$ and $x\in [\alpha ,\beta ].$ Show that, if $f{}'(x)=0$ and $f{}'(x)>0,$ then $f$ must have a local maximum at $x$ .

Ques 14.

Suppose that $f\: :[0,2]\rightarrow \mathbb{R}$ is continuous on $[0,2]$ and differentiable on $]0,2[$ and that f )0( = ,0 f )1( = ,1 f )2( = .1 (i)

Show that there exists $c_{1}\in (0,1)$ such that ${f}'(c_{1})=1.$

Show that there exists $c_{2}\in (0,1)$ such that ${f}'(c^{2})=0,$

(iii) Show that there exists $c\in (0,2)$ such that . ${f}'(c)=\frac{1}{3}.$

Ques 15.

Test the following series for convergence.

(i) $\sum_{n=1}^{\infty }n\: x^{n-1}\, ,x> 0\: .$

(ii) $\sum_{n=1}^{\infty }\left [ \sqrt{n^{4}+9}-\sqrt{n^{4}-9} \right ]$

Ques 16.

Show that $\sum_{n=1}^{\infty }(-1)^{n+1}\frac{5}{7n+2}$ is conditionally convergent.

Ques 17.

Use Cauchy’s Mean Value Theorem to prove that:

$\frac{cos\, \alpha -cos\, \beta }{sin\, \alpha-sin\, \beta }=tan\theta ,0< \alpha < \theta< \beta < \frac{\pi }{2}$

Ques 18.

Using Weiestrass M-test, show that the following series converges uniformly. $\sum_{n=1}^{\infty }n^{3}\: x^{n}\: ,x\in \left [ -\frac{1}{3},\frac{1}{3} \right ].$

Ques 19.

Use the Fundamental Theorem of Integral Calculus to evaluate the integral

$\int_{0}^{1}\left ( 2x\, sin\frac{1}{x}-cos\frac{1}{x} \right )dx.$

Ques 20.

Show that the function $f\, :\mathbb{R}\rightarrow \mathbb{R}$ defined by $f(x)=2x+7$ has an inverse by applying the inverse function theorem. Find its inverse also.

Ques 21.

Check whether the series $f(x)=2x+7\sum_{n=1}^{\infty }\frac{n^{2}x^{5}}{n^{4}+x^{3},},x\in[0,\alpha ]$ is uniformly convergent or not, wher $\alpha \in \mathbb{R}^{+}.$

Ques 22.

Show that the series $\sum_{n=1}^{\infty }\frac{sin\, n\theta }{n}$ does not converge uniformly on the interval $\left ]0,2\pi \right [$ $.$

Ques 23.

If the power series $.\sum_{n=0}^{\infty }a_{n}x^{n}$ converges uniformly in $\left ]\alpha ,\beta \right [,$ then so does $\sum_{n=0}^{\infty }a_{n}(-x)^{n}.$ True or false? Justify.