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BMTC 133: Real Analysis

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Title Name	IGNOU BMTC 133 SOLVED ASSIGNMENT
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BSCG
Course Name	Bachelor of Science
Subject Code	BMTC 133
Subject Name	Real Analysis
Year	2026
Session	-
Language	English Medium
Assignment Code	BMTC 133/Assignment-1/2026
Product Description	Assignment of BSCG (Bachelor of Science) 2026. Latest BMTC 133 2026 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU BEGC-131 (BAG) 2025-26 Assignment is for January 2026 Session: 30th September, 2026 (for December 2025 Term End Exam). Semester Wise January 2025 Session: 30th March, 2026 (for June 2026 Term End Exam). July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).
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BMTC 133 2025 - English

ASSIGNMENT

Course Code: BMTC-133

Assignment Code: BMTC-133/TMA/2025

Maximum Marks: 100

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^~qis p→q.

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

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

The function f: R→R defined by f(x) = |x-2|+|3-x is differentiable at x = 5.

2. a) Test the following series for convergence.

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

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

b) Prove that the sequence (a_n)_n∈N, where a_n = 3²/ x²+ 2^{2 ,}is Cauchy.

3. a) Show that the set:

$S=left{-frac{1}{n}+frac{1+(-1)^n}{3}:ninmathbb{N} ight}$

is not closed.

b) Test the following series for convergence:

$frac{1}{3.4}+frac{sqrt{2}}{5.6}+frac{sqrt{3}}{7.8}+ldots ext{to}infty$

4. a) Prove that a function f : S → S (where S is a finite non-empty set) is injective if it is surjective.

b) Disprove the statement:

$"(x+y)^n=x^n+y^n,forall ninmathbb{N},x,yinmathbb{Z}"$

by providing a suitable counter-example.

c) Find the supremum and infimum of the set:

$S=left{frac{1}{n-1}:ngeq 2 ight}.$

5. a) Show that [a,∞)is closed set.

b) Write the following statement, and its negation, using logical quantifiers. Also interpret its negation in words.

∃ x∈R such that x-1/3 > 0.

c) Let x and y be two real numbers such that x < y.Show that there exists an irrational number λ such that

x < λ < y.

6. a) Prove that between any two real roots of ,2 e cos2x = x there is at least one real root of e^x sin 2x = 1.

b) Let f : R → R be a function defined by:

$f(x)=egin{cases}x^3cosleft(frac{1}{2x} ight),& ext{if}x eq 0\0,& ext{if}x=0end{cases}$

Check whether f ′ is continuous on R.

7. a) Apply the Cauchy’s integral test to evaluate:

$lim_{n oinfty}left[frac{n+1}{n^2+1^2}+frac{n+2}{n^2+2^2}+frac{n+3}{n^2+3^2}+cdots+frac{1}{n} ight]$

b) For x∈ ]2,0[ and n∈N, define f_n (x) = 3 x² + 2x/n. Find the limit function ' f ' of the sequence (f_n )_n∈N , Is f continuous? Check if $int_{0}^{2}f(x)dx$ and $lim_{n oinfty}int_{0}^{2}f_{n}(x)dx$ are equal or not.

8. a) Find the radius of convergence of the series $sum a_n x^nquad ext{where}quad a_n=frac{n!}{n^n}$ .

b) $f(x)=egin{cases}frac{x+2}{x^2-4},& ext{when}x eq-2\-1,& ext{when}x=-2end{cases}$

Check whether f is uniformly continuous on [− ]1,1 or not.

c) Show that

$e^{-x}<1-x+frac{x^2}{2!},x>0.$

9. a) Show that the sequence {f_n}of functions, where

$f_n(x)=frac{n}{x+n}$

is uniformly convergent in ,0[ k],where k > .0 Show further that } { n f is not uniformly convergent in ,0[ ∞[.

b) Evaluate $int_{0}^{1}x^{2}dx$ using Riemann integration.

10. a) Find the value/s of x for which the series

$sumfrac{1.3.5...(2n-1)}{2.4.6....2n}cdotfrac{x^n}{n}$

is convergent.

b) Give one example each for the following. Justify your choice of examples.

i) A sequence which is divergent.

ii) A set which is neither open nor closed.

iii) A compact set.

iv) A set which has no limit point

BMTC 133 (January 2026 - July 2026) - ENGLISH

Assignment
(To be done after reading the course material)

Course Code: BMTC-133
Assignment Code: BMTC-133/TMA/2026
Maximum Marks: 100

1. State whether the following statements are true or false. Justify your answers with a short proof or a counterexample.

i) The negation of $p \lor (\sim q)$ is $q \to p$ .

ii) $\mathbb{Q} \setminus \mathbb{N}$ is countable.

iii) The equation $x^3 - 2x + 3 = 0$ has a real root between -2 and 1.

iv) Every increasing sequence has a convergent subsequence.

v) The function defined as
$$f(x) = \begin{cases} \frac{\sin 2x}{x}, & \text{if } x \neq 0 \\ 3, & \text{if } x = 0 \end{cases}$
has a removable discontinuity.

vi) An integrable function can have finitely many points of discontinuity.

vii) The series $\sum_{n=0}^{\infty} (-1)^n \frac{1}{2^n}$ is divergent.

viii) The function f defined by $f(x) = \sin x(1 + \cos x), \ x \in [0, 2\pi]$ , has local maximum at $x = \frac{\pi}{3}$ .

ix) Every constant function on [a, b] is Riemann Integrable.

x) The function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = |x - 2| + |x - 1|$ is differentiable at $x = 3$ .

2. (a) Using mathematical induction, prove that the sum of the squares of first 'n' natural numbers is $\frac{n(n+1)(2n+1)}{6}$ .

(b) State Cauchy’s second theorem on limits. Use it to show that:
$$\lim_{n\to\infty} \left[ \frac{n}{[(n+1)(n+2)...(2n)]^{\frac{1}{n}}} \right] = \frac{e}{4}$

(c) Check whether or not the sequence $(a_n)_{n \in \mathbb{N}}$ , defined by:
$$a_{n+1} = 2 - \frac{1}{a_n + 2} \quad \forall n \ge 1$
and $a_1 = 2$ is convergent.

3. (a) Test the convergence of the series: $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}n}{n^3 + 1}$ .
(b) Find the radius of convergence of the series $\sum a_n x^n$ , where $a_n = \frac{n!}{n^n}$ .
(c) Show that the set:
$$S = \left\{ \frac{1 + (-1)^n}{2} + \frac{1}{n} \ \middle| \ n \in \mathbb{N} \right\}$
is not closed.

4. (a) Show that the function $f : ]0, \infty[ \to \mathbb{R}$ defined as $f(x) = \frac{1}{x}$ is continuous on its domain, but not uniformly continuous there. Will the function be uniformly continuous on $[c, \infty[$ for any fixed c > 0?

(b) Examine the type of discontinuity of the function at $x = 0$ where
$$f(x) = \begin{cases} x - |x|, & \text{if } x \neq 0 \\ 2, & \text{if } x = 0. \end{cases}$

(c) Check whether the following sets are open or closed or neither:
(i) $\{3n : n \in \mathbb{N}\}$
(ii) $]1, 6[ \cup [2, 8]$

5. (a) Suppose that f and g are in R[a, b]. Then show that the function f + g is in R[a, b], and
$$\int_{a}^{b} f + g = \int_{a}^{b} f + \int_{a}^{b} g.$
(b) Discuss the convergence of the p-series $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^p}, \ p > 0$ by using the integral test.

6. (a) Prove that $\sin x < x, \ x \in [0, \frac{\pi}{2}]$ .

(b) Prove that every convergent sequence is Cauchy.

(c) Disprove the statement:
$$(x + y)^n = x^n + y^n \quad \forall n \in \mathbb{N}, \quad x, y \in \mathbb{Z}$
by providing a suitable counter-example.

7. (a) State Bolzano-Weierstrass theorem. Use it to check whether the set:
$$S = \left\{ \frac{1}{m} - \frac{1}{n} : m, n \in \mathbb{N} \right\}$
has a limit point.

(b) Give one example for each of the following. Justify.
(i) A set which is neither open or closed.
(ii) A set which has no limit point.

(c) Check whether the set of integers is countable or not.

8. (a) Let f be a differentiable function whose derivative never vanishes on [a, b]. Show that f is either strictly increasing or strictly decreasing.
(b) If an infinite series $\sum a_n$ converges, then show that $\lim_{n \to \infty} a_n = 0$ .

BMTC 133 2025 - Hindi

सत्रीय कार्य

पाठ्यक्रम कोड: बीएमटीसी-133

सत्रीय कार्य कोड: बीएमटीसी-133/त्मा/2025

अधिकतम अंक: 100

1. निम्नलिखित में से कौन-से कथन सत्य हैं या असत्य हैं? लघु उपपत्ति या प्रति-उदाहरण जो भी उचित हो, के साथ अपने उत्तरों के कारण बताइए :

i) प्रत्येक अनंत समुच्चय एक विस्तारित समुच्चय है।

ii) $pwedgesim q$ का निषेध $p ightarrow q$ है।

iii) - 1 अन्तराल ]-2,1] का सीमा बिन्दु है।

iv) फलन f के समाकलनीय होने के अनिवार्य प्रतिबंध है कि वह संतत हो।

v) $f(x)=|x-2|+|3-x|$ द्वारा परिभाषित फलन $f:mathbb{R} omathbb{R},quad x=5^circ$ पर अवकलनीय है।

2. क) निम्नलिखित श्रेणियों के अभिसरण की जाँच कीजिए :

(i) $sum_{n=1}^{infty}nx^{n-1},quad x>0$

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

ख) कॉशी अनुक्रम को परिभाषित कीजिए। सिद्ध कीजिए कि अनुक्रम $(a_n)_{ninmathbb{N}}$ , जहाँ $a_n=frac{3^2}{x^2+2^2}$ कॉशी है।

3. क) दिखाइए कि समुच्चय :

$S=left{frac{1}{n}+frac{1+(-1)^n}{3}:ninmathbb{N} ight}$

विवृत नहीं है।

ख) निम्नलिखित श्रेणी के अभिसरण की जाँच कीजिए :

$frac{1}{3cdot 4}+frac{sqrt{2}}{5cdot 6}+frac{sqrt{3}}{7cdot 8}+dotsb ext{to}infty$

4. क) सिद्ध कीजिए कि फलन फ: ऍस ऍस (जहाँ S परिमित अरिक्त समुच्चय है) एकैकी है, यदि यह आच्छादी है।

ख) एक उचित प्रति-उदाहरण देते हुए निम्नलिखित कथन को असिद्ध कीजिए :

$(x+y)^n=x^n+y^nquadforall ninmathbb{N},x,yinmathbb{Z}$

ग) समुच्चय $S=left{frac{1}{n-1}:nge 2 ight}$ का उच्चक और निम्नक ज्ञात कीजिए।

5. क) दिखाइए कि प्रत्येक [a,∞) एक विवृत समुच्चय है।

ख) निम्नलिखित कथन तथा उनके निषेध को तर्कसंगत प्रमात्रकों का प्रयोग करते हुए आप किस प्रकार प्रस्तुत करेंगे? निषेध को शब्दों में भी दीजिए।

$exists xinmathbb{R}quad ext{s.t.}quad x-frac{1}{3}>0$

ग) मान लीजिए कि ऍक्स और वाइ दो वास्तविक संख्याएँ इस प्रकार हैं कि x < y है। ऐसी एक अपरिमेय संख्या $lambda$ मालूम कीजिए जिस के लिए निम्नलिखित होता है।

x < $lambda$ < y

6. क) सिद्ध कीजिए कि e^x cos2x = 2 के किन्हीं दो वास्तविक मूलों के बीच e^x sin2x = 1 का कम से कम एक वास्तविक मूल होता है।

ख) मान लीजिए $f:mathbb{R} omathbb{R}$

$f(x)=egin{cases}x^3cosleft(frac{1}{2x} ight),& ext{if}x e 0\0,& ext{if}x=0end{cases}$

द्वारा परिभाषित फलन है। दिखाइए कि $f',mathbb{R}$ पर संतत है लेकिन x = 0 पर अवकलनीय नहीं है।

7. क) कॉशी समाकल परीक्षण द्वारा निम्नलिखित मूल्यांकन कीजिए :

$lim_{x oinfty}left[frac{n+1}{n^2+1^2}+frac{n+2}{n^2+2^2}+frac{n+3}{n^2+3^2}+dots+frac{1}{n} ight]$

ख) समुच्चय x∈ [0,2] और n∈ $mathbb{N}$ , के लिए $f_n(x)=3x^2+frac{2x}{n}$ लीजिए। अनुक्रम $(f_n)_{ninmathbb{N}}$ का ॲन् सीमा फलन 'f' ज्ञात कीजिए। क्या f संतत है? जाँच कीजिए कि $int_{0}^{2}f(x),dx$ और $lim_{n oinfty}int_0^2 f_n(x),dx$ समान हैं या नहीं।

8. a) त्रिज्या $sum a_n x^n$ जहाँ $a_n=frac{n!}{n^n}$ की अभिसरण त्रिज्या ज्ञात कीजिए I

ख) मान लीजिए $f(x)=egin{cases}frac{x+2}{x^2-4},& ext{where},x e-2\-1,& ext{where}:x=-2end{cases}$

जाँच की f, [-1,1] पर एकसमान सांतत्य है या नहीं।

ग) सिद्ध कीजिए :

$e^{-x}<1-x+frac{x^2}{2!},quad x>0$

9. क) सिद्ध कीजिए कि फलनों का अनुक्रम {f_n} जहाँ

$f_n(x)=frac{n}{x+n}$

[0,k] पर एकसमानतः अभिसरित है जहाँ k> 0 है। आगे सिद्ध कीजिए कि {f_n}, [0,∞ [ पर एकसमानतः अभिसरित नहीं है।

ख) रीमान समाकलन से $int_{0}^{1}x^2,dx$ परिकलित कीजिए।

10. क) ऍक्स के वे मान ज्ञात कीजिए जिन के लिए श्रेणी

$sum_{n=1}^{infty}frac{1cdot 3cdot 5cdots(2n-1)}{2cdot 4cdot 6cdots 2n}cdotfrac{x^n}{n}$

अभिसरित होती है।

ख) निम्नलिखित के एक-एक उदाहरण दीजिए। अपने चयन की पुष्टि कीजिए।

i) एक अनुक्रम जो अपसारी है।

ii) एक समुच्चय जो विवृत है लेकिन संवृत नहीं है।

iii) संहत समुच्चय I

iv) एक समुच्चय जिसका कोई सीमा बिन्दु नहीं है।

BMTC 133 (January 2026 - July 2026) - HINDI

सत्रीय कार्य
(पाठ्य सामग्री पढ़ने के बाद ही इसे हल करें।)

पाठ्यक्रम कोड: BMTC-133

सत्रीय कार्य कोड: BMTC-133/TMA/2026

अधिकतम अंक: 100

1. जाँच कीजिए कि निम्नलिखित कथन सत्य हैं अथवा असत्य। अपने उत्तर की लघु-व्याख्या या प्रति-उदाहरण द्वारा पुष्टि कीजिए। ()

(i) $p \lor (\sim q)$ का निषेध $q \to p$ है।

(ii) $\mathbb{Q} \setminus \mathbb{N}$ गणनीय हैं।

(iii) समीकरण $x^3 - 2x + 3 = 0$ का एक वास्तविक मूल -2 और 1 के बीच है।

(iv) प्रत्येक वर्धमान अनुक्रम का एक अभिसारी उप-अनुक्रम होता है।

(v) फलन $f(x)=egin{cases}dfrac{sin 2x}{x},&x e 0\3,&x=0end{cases}$ में एक अपनेय असांतत्य है।

(vi) एक समाकलनीय फलन में असांतत्य के सीमित बिंदु होते है।

(vii) श्रेणी $\sum_{n=0}^{\infty} (-1)^n \frac{1}{2^n}$ अपसारी है।

(viii) फलन f, जो $f(x) = \sin x (1 + \cos x)$ , द्वारा परिभाषित है तथा $x \in [0, 2\pi]$ पर परिभाषित है, का स्थानीय अधिकतम $x = \frac{\pi}{3}$ पर हैं।

(ix) अंतराल [a, b] पर परिभाषित प्रत्येक अचर फलन रीमान समाकलनीय होता है।

(x) फलन $f: \mathbb{R} \to \mathbb{R}$ जो $f(x) = |x - 2| + |x - 1|$ द्वारा परिभाषित है, $x = 3$ पर अवकलनीय है।

2. (a) गणितीय आगमन का प्रयोग करके सिद्ध कीजिए कि प्रथम 'n' प्राकृतिक संख्याओं के वर्गों का योगफल $\frac{n(n+1)(2n+1)}{6}$ होता है।

(b) सीमाओं पर कौशी का द्वितीय प्रमेय लिखिए। इसे यह प्रदर्शित करने के लिए प्रयोग कीजिए कि $\lim_{n \to \infty} \left( \frac{n}{\{(n+1)(n+2)...(2n)\}^{\frac{1}{n}}} \right) = \frac{e}{4}$ हो।

(c) जाँच कीजिए कि अनुक्रम $(a_n)_{n \in \mathbb{N}}$ , जिसे इस प्रकार से परिभाषित किया जाता है:
$a_{n+1} = 2 - \frac{1}{a_n + 2} \quad \forall n \geq 1$ , और $a_n = 2$ .
अभिसारी है या नहीं।

3. (a) श्रेणी $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}n}{n^3+1}$ के अभिसरण की जाँच कीजिए।

(b) श्रेणी $\sum_{n=1}^{\infty} a_nx^n$ , की अभिसरण त्रिज्या ज्ञात कीजिए। जहाँ $a_n = \frac{n!}{n^n}$ है।

(c) सिद्ध कीजिए कि यह समुच्चय $S = \left\{ \frac{1+(-1)^n}{2} + \frac{1}{n} \mid n \in \mathbb{N} \right\}$ संवृत नहीं है।

4. (a) दिखाइए कि फलन $f : ]0, \infty[ \to \mathbb{R}$ , जो $f(x) = \frac{1}{x}$ द्वारा परिभाषित है, अपने प्रांत पर सतत है, परंतु वहाँ समान रूप से सतत नहीं है। क्या यह फलन किसी भी निश्चित c > 0 के लिए अंतराल $[c, \infty[$ पर एकसमान सतत होगा?

(b) फलन f की $x = 0$ पर असांतत्य के प्रकार की जाँच कीजिए, जहाँ
$f(x)=egin{cases}x-|x|,&x eq 0\2,&x=0end{cases}$ है।

(c) जाँच कीजिए कि निम्नलिखित समुच्चय विवृत हैं या संवृत हैं अथवा न तो विवृत हैं और न ही संवृत।

i. $\{3n : n \in \mathbb{N}\}$

ii. $]1, 6[ \cup [2, 8]$
5. (a) मान लीजिए कि f तथा g, R[a, b] में हैं। तब दिखाइए कि फलन f + g भी R[a, b] में हैं, तथा $\int_a^b (f + g) = \int_a^b f + \int_a^b g$ .
(b) समाकल परीक्षण का प्रयोग करके p − श्रेणी $sum_{n=1}^{infty}frac{1}{n^p}$ , जहाँ p > 0, के अभिसरण पर चर्चा कीजिए।

6. (a) सिद्ध कीजिए कि $\sin x < x$ , जहाँ x अंतराल $[0, \frac{\pi}{2}]$ में है।

(b) सिद्ध कीजिए कि प्रत्येक अभिसारी अनुक्रम कौशी होता है।

(c) उपयुक्त प्रति उदाहरण देकर इस कथन को असिद्ध कीजिए:

प्रत्येक $n \in \mathbb{N}$ तथा $x, y \in \mathbb{Z}$ के लिए $(x + y)^n = x^n + y^n$ है।

7. (a) बोलजनों-वेयरस्ट्रास प्रमेय लिखिए। इसका प्रयोग करके जाँच कीजिए कि समुच्चय $S = \{ \frac{1}{m} - \frac{1}{n} : m, n \in N \}$ का कोई सीमा-बिंदु है या नहीं।

(b) निम्नलिखित में से प्रत्येक के लिए एक - एक उदाहरण दीजिए। अपने उत्तर का कारण बताइए।

i. एक ऐसा समुच्चय जो न तो विवृत है और न संवृत है।

ii. एक ऐसा समुच्चय जिसका कोई सीमा बिंदु नहीं है।

(c) जाँच कीजिए कि पूर्णांकों का समुच्चय गणनीय है या नहीं।

8. (a) मान लीजिए कि f एक अवकलनीय फलन है, जिसका अवकलज अंतराल [a, b] पर शून्य नहीं होता। तब दिखाइए कि f या तो सर्वथा वर्धमान है अथवा सर्वथा ह्रासमान है।

(b) यदि अनंत श्रेणी $\sum a_n$ अभिसारी है, तो दिखाइए कि $\lim_{n \to \infty} a_n = 0$ होती है।

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