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MTE 9: Real Analysis

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Title Name	IGNOU MTE 9 SOLVED ASSIGNMENT
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BSC
Course Name	Bachelor in Science
Subject Code	MTE 9
Subject Name	Real Analysis
Year	2025
Session	-
Language	English Medium
Assignment Code	MTE 9/Assignment-1/2025
Product Description	Assignment of BSC (Bachelor in Science) 2025. Latest MTE 09 2025 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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MTE 9 2025 - English

ASSIGNMENT

Course Code: MTE-09

Assignment Code: MTE-09/TMA/2025

Maximum When: 100

1. Are the following statements true or false? Give reasons for tour answers.

a) -2 is a limit point of the interval 1-3,2].

b) The series $frac{1}{2}-frac{1}{6}+frac{1}{10}-frac{1}{14}+cdots$ is divergent.

The function, f(x)=sin² x is uniformly continuous in the interval [0, π).

d) Every continuous function is differentiable.

e) The function f defined on R by $f(x)=egin{cases}0,& ext{x is rational}\2,& ext{x is irrational}end{cases}$

is irrational Is integrable in the interval [2,3].

2) Prove that the union of two closed sets is a closed set. Give an example to show that union of an infinite number of closed sets need not be a closed set.

b) Examine the function f : R→R defined by

$f(x)=egin{cases}frac{1}{6}(x+1)^3&x e 0\frac{5}{6}&x=0end{cases}$

for continuity on R. If it is not continuous at any point of R, find the nature of discontinuity there. (x+1)x+0 9-59-1

Find lim $lim_{x o 0}frac{1-cos^2 x}{x^2sin^2 x}$

3) Using the principle of mathematical induction, prove that $frac{n^5}{5}+frac{n^3}{3}+frac{7n}{15}$ is natural numbuer. $forall ninmathbb{N}.$

b) Show that there is no real number, & for which the equation, $x^4-3x^2+k=0$ has two distinct roots in the interval [2,3].

c) Let :[-3,3]→ R be defined by $f(x)=5(x)+x^3,$ where [x] denotes the greatest integer ≤x. Show that this function is integrable.

4. a) Prove that the function $f$ defined by

$f(x)=egin{cases}2,& ext{if}x ext{is irrational}\-2,& ext{if}x ext{is rational}end{cases}$ l is discontinuous, $forall xinmathbb{R},$ , using the sequential definition of continuity.

b) Examine the convergence of the following series

(i) $frac{3 imes4}{5^2}+frac{5 imes6}{7^2}+frac{7 imes8}{9^2}+cdots$

(ii) $1+4x+4^2 x^2+4^3 x^3+cdots(x>0)$

c) Prove that the set of integers is countable.

5. Prove that

$lim_{n ightarrowinfty}left[frac{1}{sqrt{2n-1^2}}+frac{1}{sqrt{4n-2^2}}+frac{1}{sqrt{6n-3^2}}+cdots+frac{1}{n} ight]=frac{pi}{2}$

b) Prove that the sequence $left{frac{a_n}{n} ight}$ is convergent where ( $alpha$ _n)is a bounded sequence.

c) Prove that continuous function of a continuous function is continuous.

6.Examine the function, $f(x)=(x+1)^3(x-3)^2$ for extreme values.

b) Show that the series $sum_{n=1}^{infty}frac{x}{1+n^2x^2}$ is uniformly convergent in $[alpha,1]$ for any a>0.

c) Give an example of an infinite set with finite number of limit points, with proper justification.

7. a) Show that

(i) $lim_{x ightarrowinfty}left(frac{x-3}{x-1} ight)^x=frac{1}{e^2}$

(ii) $lim_{x ightarrowfrac{5}{3}}frac{1}{(3x+5)^2}=infty$

b) For the function, $f(x)=x^2-2$ defined over [1,5], verify: $L(P,f)le U(-P,f)$ where P is the partition which divides [1.5] into four equal intervals.

c) Let (a) be a sequence defined as $a_1=3,a_{n+1}=frac{1}{5}a_n$ show that (a_n) converges

8.a) Using the sequential definition of the continuity, prove that the function $f$ defined

by: $f(x)=egin{cases}3,& ext{if}x ext{is irrational}\-3,& ext{if}x ext{is rational}end{cases}$ is discontinuous at each real number.

b) Show that on the curve, $y=3x^2-7x+6,$ the chord joining the points whose abscissa are x=1 and x=2. is parallel to the tangent at the whose abscissa is $a xfrac{3}{2}.$

c) Give an example of a series ∑ $a_{n}$ such that ∑ $a_{n}$ is not convergent but the $(a_{n)$ sequence converges to 0.

9. a) Test the series:

$sum_{n=1}^{infty}(-1)^{n-1}frac{sin nx}{nsqrt{n}}$ for absolute and conditional convergence.

b) Check whether the function $f$ given by:

$f(x)=(x-4)^3(x+1)^2$ has local maxima and local minima.

c) Check, whether the collection $G'=left{left]frac{1}{n+2},frac{1}{n} ight[:ninmathbb{N} ight}$ given by: open cover of $]0,1[$

10. a)State Bonnet's mean value theorem for integrals. Apply it to show that:

$left|int_{3}^{5}frac{cos x}{x}dx ight|lefrac{2}{3}$

b) Show that the sequence (a), where $a_nfrac{n}{n^2+4}$ is monotonic. Is $(a_n)a$ Cauchy sequence Justify your answer.

c) Check whether the intervals [2.5] and [7,10] are equivalent or not.

MTE 9 2025 - Hindi

सत्रीय कार्य

पाठ्यक्रम कोड: एम. टी. ई. - 09

सत्रीय कार्य कोड एम. टी. ई.09/ टी एम ए / 2025

अधिकतम अंक: 100

1. बताइए निम्नलिखित कथन सत्य हैं या असत्य। अपने उत्तरों के कारण बताइए।

क) - 2 अंतराल ]-3,2] का सीमा बिन्दु है।

ख) श्रेणी $frac{1}{2}-frac{1}{6}+frac{1}{10}-frac{1}{14}+cdots$ अपसारी है।

ग) फलन f (x) = sin²x अंतराल [0, $pi$ ] पर एकसमानतः सतत है।

घ) प्रत्येक सतत फलन अवकलनीय है।

ड.) $f(x)=egin{cases}0,& ext{x is rational}\2,& ext{x is irrational}end{cases}$ द्वारा $mathbb{R}$ पर परिभाषित फलन ƒ अंतराल [2, 3] में समाकलनीय है।

2) क) सिद्ध कीजिए कि दो संवृत समुच्चयों का सम्मिलन संवृत समुच्चय है। यह दिखाने के लिए एक उदाहरण दीजिए कि संवृत समुच्चयों की परिमित संख्या का सम्मिलन संवृत समुच्चय नहीं भी हो सकता है।

ख) R पर सातत्य के लिए

$f(x)=egin{cases}frac{1}{6}(x+1)^3&x e 0\frac{5}{6}&x=0end{cases}$

द्वारा परिभाषित फलन f: IR IR की जाँच कीजिए। यदि R के किसी बिन्दु पर सतत नहीं है तब असातत्य का प्रकार ज्ञात कीजिए

ग) $lim_{x o 0}frac{1-cos^2 x}{x^2sin x^2}$ ज्ञात कीजिए।

3) क) गणितीय आगम नियम द्वारा सिद्ध कीजिए कि $frac{n^5}{5}+frac{n^3}{3}+frac{7n}{15}$ एक प्राकृत संख्या है, $forall ninmathbb{N}.$

ख) दिखाइए कि ऐसी कोई वास्तविक संख्या नहीं है जिसके लिए समीकरण x⁴-3x²+k=0 के अंतराल [2,3] में दो अलग-अलग मूल होते हैं।

ग) मान लीजिए $f:[-3,3] omathbb{R},quad f(x)=5(x)+x^3$ द्वारा परिभाषित है, जहाँ [x] महत्तम पूर्णांक $le x$ को निरूपित करता है। दिखाइए कि यह फलन समाकलनीय है।

4. क) सातत्य की अनुक्रमिक परिभाषा द्वारा सिद्ध कीजिए कि

$f(x)=egin{cases}2,& ext{if x is irratioanl}\-2,& ext{if x is rational}end{cases}$

द्वारा परिभाषित फलन f असतत है, $forall xinmathbb{R}$ I

ख) निम्नलिखित श्रेणी के अभिसरण की जाँच कीजिए:

i) $frac{3 imes 4}{5^2}+frac{5 imes 6}{7^2}+frac{7 imes 8}{9^2}+cdots$

ii) $1+4x+4^2 x^2+4^3 x^3+cdotsquad(x>0)$

ग) सिद्ध कीजिए कि पूर्णांकों का समुच्चय गणनीय है।

5. क) सिद्ध कीजिए कि

$lim_{n ightarrowinfty}left[frac{1}{sqrt{2n-1^2}}+frac{1}{sqrt{4n-2^2}}+frac{1}{sqrt{6n-3^2}}+cdots+frac{1}{sqrt{2n(n)-n^2}} ight]=frac{pi}{2}$

ख) सिद्ध कीजिए कि अनुक्रम $left{frac{a_n}{n} ight}$ अभिसारी। है जहाँ {a_n} परिबद्ध अनुक्रम है।

ग) सिद्ध कीजिए कि संतत फलन का संतत फलन संतत होता है।

6. क) चरम मानों के लिए फलन f(x) = (x+1)³ (x-3)² की जाँच कीजिए।

ख) दिखाइए कि किसी भी a > 0 श्रेणी $sum_{n=1}^{infty}frac{x}{1+n^{2}x^{2}},quad[alpha,1]$ में एकसमानतः अभिसारी है।

ग) उचित पुष्टि के साथ सीमा बिन्दुओं की परिमित संख्या वाले एक अपरिमित समुच्चय का उदाहरण दीजिए।

7. क) दिखाइए कि

i) $lim_{x ightarrowinfty}left(frac{x-3}{x-1} ight)^{x}=frac{1}{e^2}$

ii) $lim_{x ightarrowfrac{5}{3}}frac{1}{(3x+5)^{2}}=infty$

ख) [1,5] पर परिभाषित फलन f(x) = x²-2 के लिए सत्यापित कीजिए L(P, f) ≤U(-P, f) जहाँ P एक विभाजन है जो [1,5] को चार बराबर अंतरालों में विभाजित करता है।

ग) मान लीजिए $a_1=3,quad a_{n+1}=frac{1}{5}a_n$ के रूप में परिभाषित अनुक्रम {a_n} है। दिखाइए कि {a_n} शून्य तक अभिसरण करता है।

8. क) फलन के सांतत्य की अनुक्रमिक परिभाषा का प्रयोग करते हुए सिद्ध कीजिए कि

$f(x)=egin{cases}3,& ext{if},,x ext{,,,is irrational}\-3,& ext{if},,x ext{,,is rational}end{cases}$

द्वारा परिभाषित फलन f, प्रत्येक वास्तविक संख्या पर असंतत है।

ख) दिखाइए कि वक्र y = 3x²-7x+6, पर बिन्दुओं को, जिनकी भुज x = 1 और x = 2, हैं, मिलाने वाली जीवा उस बिन्दु पर खींची गई स्पर्श रेखा के समान्तर होती है जिसकी भुजा $x=frac{3}{2}.$

ग) ऐसी श्रेणी $sum a_n$ का उदाहरण दीजिए जिसके लिए $sum a_n$ , अभिसारी नहीं है लेकिन अनुक्रम (a_n) 0 की ओर अभिसरण करता है।

9. क) निरपेक्ष और सप्रतिबंध अभिसरण के लिए श्रेणी

$sum_{n=1}^{infty}(-1)^{n-1}frac{sin nx}{nsqrt{n}}$

की जाँच कीजिए।

ख) जाँच कीजिए :

f (x) = (x-s)³ (x+1)²

द्वारा दिए गए फलन f का स्थानीय उच्चिष्ठ और स्थानीय निम्निष्ठ होता है या नहीं।

ग) जाँच कीजिए कि :

$G'=left{left]frac{1}{n+2},frac{1}{n} ight[:ninmathbb{N} ight}$

द्वारा दिया गया संग्रह G, ]0,1[. का विवृत आवरक है या नहीं।

10. क) समाकलों के लिए बोनट के माध्य मान प्रमेय का कथन दीजिए

$left|int_{3}^{5}frac{cos x}{x}dx ight|leqfrac{2}{3}$

दिखाने के लिए इसे लागू कीजिए।

ख) दिखाइए कि अनुक्रम (a_n), एकदिष्ट है, जहाँ $a_nfrac{n}{n^2+4}1$ उत्तर की पुष्टि कीजिए। । क्या (a_n) कॉशी अनुक्रम है? अपने

ग) जाँच कीजिए कि अंतराल [2,5] और [7,10] तुल्य हैं या नहीं।

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