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MATHEMATICS-I. CONTENTS Ordinary Differential Equations of First Order and First Degree Ordinary Differential Equations of First Order and First Degree.

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Presentation on theme: "MATHEMATICS-I. CONTENTS Ordinary Differential Equations of First Order and First Degree Ordinary Differential Equations of First Order and First Degree."— Presentation transcript:

1 MATHEMATICS-I

2 CONTENTS Ordinary Differential Equations of First Order and First Degree Ordinary Differential Equations of First Order and First Degree Linear Differential Equations of Second and Higher Order Linear Differential Equations of Second and Higher Order Mean Value Theorems Mean Value Theorems Functions of Several Variables Functions of Several Variables Curvature, Evolutes and Envelopes Curvature, Evolutes and Envelopes Curve Tracing Curve Tracing Applications of Integration Applications of Integration Multiple Integrals Multiple Integrals Series and Sequences Series and Sequences Vector Differentiation and Vector Operators Vector Differentiation and Vector Operators Vector Integration Vector Integration Vector Integral Theorems Vector Integral Theorems Laplace transforms Laplace transforms

3 TEXT BOOKS A text book of Engineering Mathematics, Vol-I T.K.V.Iyengar, B.Krishna Gandhi and Others, S.Chand & Company A text book of Engineering Mathematics, Vol-I T.K.V.Iyengar, B.Krishna Gandhi and Others, S.Chand & Company A text book of Engineering Mathematics, C.Sankaraiah, V.G.S.Book Links A text book of Engineering Mathematics, C.Sankaraiah, V.G.S.Book Links A text book of Engineering Mathematics, Shahnaz A Bathul, Right Publishers A text book of Engineering Mathematics, Shahnaz A Bathul, Right Publishers A text book of Engineering Mathematics, P.Nageshwara Rao, Y.Narasimhulu & N.Prabhakar Rao, Deepthi Publications A text book of Engineering Mathematics, P.Nageshwara Rao, Y.Narasimhulu & N.Prabhakar Rao, Deepthi Publications

4 REFERENCES A text book of Engineering Mathematics, B.V.Raman, Tata Mc Graw Hill A text book of Engineering Mathematics, B.V.Raman, Tata Mc Graw Hill Advanced Engineering Mathematics, Irvin Kreyszig, Wiley India Pvt. Ltd. Advanced Engineering Mathematics, Irvin Kreyszig, Wiley India Pvt. Ltd. A text Book of Engineering Mathematics, Thamson Book collection A text Book of Engineering Mathematics, Thamson Book collection

5 UNIT-VI SERIES AND SEQUENCES

6 UNIT HEADER Name of the Course: B.Tech Code No:07A1BS02 Year/Branch: I Year CSE,IT,ECE,EEE,ME,CIVIL,AERO Unit No: VI No. of slides:21

7 S. No. ModuleLectureNo. PPT Slide No. 1 Introduction, Comparison test and Auxiliary series L DAlemberts, Cauchys, Integral, Raabes and Logarithmic tests L Alternating series, Absolute and Conditional convergence L UNIT INDEX UNIT-VI

8 Lecture-1 SEQUENCE A Sequence of real numbers is a set of numbers arranged in a well defined order. Thus for each positive integer there is associated a numbr of the sequence. A function s:Z + R is called a SEQUENCE of real numbers. A Sequence of real numbers is a set of numbers arranged in a well defined order. Thus for each positive integer there is associated a numbr of the sequence. A function s:Z + R is called a SEQUENCE of real numbers. Example 1:1,2,3,…….. Example 1:1,2,3,…….. Example 2:1,1/2,1/3,………… Example 2:1,1/2,1/3,…………

9 CONVERGENT,DIVERGENT, OSCILLATORY SEQUENCE If limit of s n =l, then we say that the sequence {s n } converges to l. If limit of s n =l, then we say that the sequence {s n } converges to l. If limit of s n =+ or - then we say that the sequence {s n } diverges to l. If limit of s n =+ or - then we say that the sequence {s n } diverges to l. If sequence is neither convergent nor divergent then such sequence is known as an Oscillatory sequence. If sequence is neither convergent nor divergent then such sequence is known as an Oscillatory sequence.

10 Lecture-2 COMPARISON TEST If Σu n and Σv n are two series of positive terms and limit of u n /v n = l0, then the series Σu n and Σv n both converge or both diverge. If Σu n and Σv n are two series of positive terms and limit of u n /v n = l0, then the series Σu n and Σv n both converge or both diverge. Example 1:By comparison test, the series Example 1:By comparison test, the series (2n-1)/n(n+1)(n+2) is convergent (2n-1)/n(n+1)(n+2) is convergent Example 2: By comparison test, the series (3n+1)/n(n+2) is divergent Example 2: By comparison test, the series (3n+1)/n(n+2) is divergent

11 Lecture-3 AUXILIARY SERIES The series Σ1/n p converges if p>1 and diverges otherwise. The series Σ1/n p converges if p>1 and diverges otherwise. Example 1: By Auxiliary series test the series 1/n is divergent since p=1 Example 1: By Auxiliary series test the series 1/n is divergent since p=1 Example 2: By Auxiliary series test the series 1/n 3/2 is convergent since p=3/2>1 Example 2: By Auxiliary series test the series 1/n 3/2 is convergent since p=3/2>1 Example 3: By Auxiliary series test the series 1/n 1/2 is divergent since p=1/2<1 Example 3: By Auxiliary series test the series 1/n 1/2 is divergent since p=1/2<1

12 Lecture-4 DALEMBERTS RATIO TEST If Σu n is a series of positive terms such that limit u n /u n+1 = l then i) Σu n converges if l>1, (ii) Σu n diverges if l 1, (ii) Σu n diverges if l<1, (iii) the test fails to decide the nature of the series, if l=1. Example : By DAlemberts ratio test the series 1.3.5….(2n-1)/2.4.6…..(2n) x n-1 is convergent if x>1 and divergent if x 1 and divergent if x<1 or x=1

13 Lecture-5 CAUCHYS ROOT TEST If Σu n is a series of positive terms such that limit u n 1/n =l then (a) Σu n converges if l 1 and (c)the test fails to decide the nature if l=1. If Σu n is a series of positive terms such that limit u n 1/n =l then (a) Σu n converges if l 1 and (c)the test fails to decide the nature if l=1. Example: By Cauchys root test the series [(n+1)/(n+2) x] n is convergnt if x 1 or x=1. Example: By Cauchys root test the series [(n+1)/(n+2) x] n is convergnt if x 1 or x=1.

14 Lecture-6 INTEGRAL TEST Let f be a non-negative decresing function of [1,). Then the series Σu n and the improper integral of f(x) between the limits 1 and converge or diverge together. Let f be a non-negative decresing function of [1,). Then the series Σu n and the improper integral of f(x) between the limits 1 and converge or diverge together. Example 1: By Integral test the series 1/(n 2 +1) is convergent. Example 1: By Integral test the series 1/(n 2 +1) is convergent. Example 2: By Integral test the series 2n 3 /(n 4 +3) is divergent. Example 2: By Integral test the series 2n 3 /(n 4 +3) is divergent.

15 Lecture-7 RAABES TEST Let Σu n be a series of positive terms and let limit n[u n /u n+1 – 1]=l. Then (a) if l>1, Σu n converges (b) if l 1, Σu n converges (b) if l<1, Σu n diverges (c) the test fails when l=1. Example: By Raabes test the series 4.7….(3n+1)/1.2…..n x n is convergent if x 1/3 or x=1/3 Example: By Raabes test the series 4.7….(3n+1)/1.2…..n x n is convergent if x 1/3 or x=1/3

16 Lecture-8 LOGARITHMIC TEST If Σu n is a series of positive terms such that limit n log[u n /u n+1 ]=l, then (a) Σu n converges if l>1 (b) Σu n diverges if l 1 (b) Σu n diverges if l<1 (c)the test fails when l=1. Example: By logarithmic test the series 1+x/2+2!/3 2 x 2 +….. is convergent if x e or x=e Example: By logarithmic test the series 1+x/2+2!/3 2 x 2 +….. is convergent if x e or x=e

17 Lecture-9 DEMORGANS AND BERTRANDS TEST Let Σu n be a series of positive terms and let limit[{n(u n /u n+1 – 1)-1}logn]=l then i)Σu n converges for l>1 and ii) diverges for l 1 and ii) diverges for l<1. Example: By Demorgans and Bertrands test the series / / /5 2 +…. is divergent Example: By Demorgans and Bertrands test the series / / /5 2 +…. is divergent

18 Lecture-10 ALTERNATING SERIES A series whose terms are alternatively positive and negativ is called an alternating series. An alternating series may be written as u 1 – u 2 + u 3 -….+(-1) n-1 u n +…… A series whose terms are alternatively positive and negativ is called an alternating series. An alternating series may be written as u 1 – u 2 + u 3 -….+(-1) n-1 u n +…… Example 1:1-1/2+1/3-1/4+….is an alternating series. Example 1:1-1/2+1/3-1/4+….is an alternating series. Example 2:(-1) n-1 n/logn is an alternating series Example 2:(-1) n-1 n/logn is an alternating series

19 Lecture-11 LEIBNITZS TEST If {u n } is a sequence of positive terms such that (a)u 1 u 2 …. u n u n+1 …… (b)limit u n =0 then the alternating series is convergent. If {u n } is a sequence of positive terms such that (a)u 1 u 2 …. u n u n+1 …… (b)limit u n =0 then the alternating series is convergent. Example 1: By Leibnitzs test the series (-1) n /n! is convergent. Example 1: By Leibnitzs test the series (-1) n /n! is convergent. Example 2: By Leibnitzs test the series (-1) n /(n 2 +1) is convergent Example 2: By Leibnitzs test the series (-1) n /(n 2 +1) is convergent

20 Lecture-12 ABSOLUTE CONVERGENCE Consider a series Σu n where u n s are positive or negative. The series Σu n is said to be absolutely convergent if Σ|u n | is convergent. Consider a series Σu n where u n s are positive or negative. The series Σu n is said to be absolutely convergent if Σ|u n | is convergent. Example 1: The series (-1) n logn/n 2 is absolute convergence. Example 1: The series (-1) n logn/n 2 is absolute convergence. Example 2: The series (-1) n (2n+1)/n(n+1)(2n+3) is absolute convergence. Example 2: The series (-1) n (2n+1)/n(n+1)(2n+3) is absolute convergence.

21 Lecture-13 CONDITIONALLY CONVERGENT SERIES If Σu n converges and Σ|u n | diverges, then we say that Σu n converges conditionally or converges non-absolutely or semi-convergent. If Σu n converges and Σ|u n | diverges, then we say that Σu n converges conditionally or converges non-absolutely or semi-convergent. Example: The series (-1) n (2n+3)/(2n+1)(4n+3) is conditional convergence. Example: The series (-1) n (2n+3)/(2n+1)(4n+3) is conditional convergence.


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