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MATHEMATICS-I

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

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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, C.Sankaraiah, V.G.S.Book Links 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

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

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UNIT-VI SERIES AND SEQUENCES

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**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

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UNIT INDEX UNIT-VI S. No. Module Lecture No. PPT Slide No. 1 Introduction, Comparison test and Auxiliary series L1-5 8-11 2 D’Alembert’s, Cauchy’s, Integral, Raabe’s and Logarithmic tests L6-10 12-16 3 Alternating series, Absolute and Conditional convergence L11-13 17-21

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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. Example 1:1,2,3,…….. Example 2:1,1/2,1/3,…………

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**CONVERGENT,DIVERGENT, OSCILLATORY SEQUENCE**

If limit of sn=l, then we say that the sequence {sn} converges to l. If limit of sn=+∞ or -∞ then we say that the sequence {sn} diverges to l. If sequence is neither convergent nor divergent then such sequence is known as an Oscillatory sequence.

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**Lecture-2 COMPARISON TEST**

If Σun and Σvn are two series of positive terms and limit of un/vn = l≠0, then the series Σun and Σvn both converge or both diverge. Example 1:By comparison test, the series ∑(2n-1)/n(n+1)(n+2) is convergent Example 2: By comparison test, the series ∑(3n+1)/n(n+2) is divergent

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**Lecture-3 AUXILIARY SERIES**

The series Σ1/np converges if p>1 and diverges otherwise. Example 1: By Auxiliary series test the series ∑1/n is divergent since p=1 Example 2: By Auxiliary series test the series ∑1/n3/2 is convergent since p=3/2>1 Example 3: By Auxiliary series test the series ∑1/n1/2 is divergent since p=1/2<1

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**Lecture-4 D’ALEMBERT’S RATIO TEST**

If Σun is a series of positive terms such that limit un/un+1 = l then i) Σun converges if l>1, (ii) Σun diverges if l<1, (iii) the test fails to decide the nature of the series, if l=1. Example : By D’Alembert’s ratio test the series ∑1.3.5….(2n-1)/2.4.6…..(2n) xn-1 is convergent if x>1 and divergent if x<1 or x=1

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**Lecture-5 CAUCHY’S ROOT TEST**

If Σun is a series of positive terms such that limit un1/n =l then (a) Σun converges if l<1, (b) Σun diverges if l>1 and (c)the test fails to decide the nature if l=1. Example: By Cauchy’s root test the series ∑[(n+1)/(n+2) x]n is convergnt if x<1 and divergent if x>1 or x=1.

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**Lecture-6 INTEGRAL TEST**

Let f be a non-negative decresing function of [1,∞). Then the series Σun and the improper integral of f(x) between the limits 1 and ∞ converge or diverge together. Example 1: By Integral test the series ∑1/(n2+1) is convergent. Example 2: By Integral test the series ∑2n3/(n4+3) is divergent.

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Lecture-7 RAABE’S TEST Let Σun be a series of positive terms and let limit n[un/un+1 – 1]=l. Then (a) if l>1, Σun converges (b) if l<1, Σun diverges (c) the test fails when l=1. Example: By Raabe’s test the series ∑4.7….(3n+1)/1.2…..n xn is convergent if x<1/3 and divergent if x>1/3 or x=1/3

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**Lecture-8 LOGARITHMIC TEST**

If Σun is a series of positive terms such that limit n log[un/un+1]=l, then (a) Σun converges if l> (b) Σun diverges if l< (c)the test fails when l=1. Example: By logarithmic test the series 1+x/2+2!/32x2+….. is convergent if x<e and divergent if x>e or x=e

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**Lecture-9 DEMORGAN’S AND BERTRAND’S TEST**

Let Σun be a series of positive terms and let limit[{n(un/un+1 – 1)-1}logn]=l then i)Σun converges for l>1 and ii) diverges for l<1. Example: By Demorgan’s and Bertrand’s test the series 1+22/32+22/32.42/52+…. is divergent

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**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 u1 – u2 + u3 -….+(-1)n-1un+…… Example 1:1-1/2+1/3-1/4+….is an alternating series. Example 2:∑(-1)n-1 n/logn is an alternating series

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**Lecture-11 LEIBNITZ’S TEST**

If {un} is a sequence of positive terms such that (a)u1≥u2 ≥…. ≥un ≥un+1 ≥…… (b)limit un=0 then the alternating series is convergent. Example 1: By Leibnitz’s test the series ∑(-1)n/n! is convergent. Example 2: By Leibnitz’s test the series ∑(-1)n/(n2+1) is convergent

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**Lecture-12 ABSOLUTE CONVERGENCE**

Consider a series Σun where un’s are positive or negative. The series Σun is said to be absolutely convergent if Σ|un| is convergent. Example 1: The series ∑(-1)n logn/n2 is absolute convergence. Example 2: The series ∑(-1)n (2n+1)/n(n+1)(2n+3) is absolute convergence.

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**Lecture-13 CONDITIONALLY CONVERGENT SERIES**

If Σun converges and Σ|un| diverges, then we say that Σun converges conditionally or converges non-absolutely or semi-convergent. Example: The series ∑(-1)n (2n+3)/(2n+1)(4n+3) is conditional convergence.

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