MATHEMATICS-IMATHEMATICS-I

CONTENTSCONTENTS Ordinary Differential Equations of First Order and First DegreeOrdinary Differential Equations of First Order and First Degree Linear Differential Equations of Second and Higher OrderLinear Differential Equations of Second and Higher Order Mean Value TheoremsMean Value Theorems Functions of Several VariablesFunctions of Several Variables Curvature, Evolutes and EnvelopesCurvature, Evolutes and Envelopes Curve TracingCurve Tracing Applications of IntegrationApplications of Integration Multiple IntegralsMultiple Integrals Series and SequencesSeries and Sequences Vector Differentiation and Vector OperatorsVector Differentiation and Vector Operators Vector IntegrationVector Integration Vector Integral TheoremsVector Integral Theorems Laplace transformsLaplace transforms

TEXT BOOKSTEXT BOOKS A text book of Engineering Mathematics, Vol-I A text book of Engineering Mathematics, Vol-I

T.K.V.Iyengar, B.Krishna Gandhi and Others, T.K.V.Iyengar, B.Krishna Gandhi and Others, S.Chand & CompanyS.Chand & Company

A text book of Engineering Mathematics, A text book of Engineering Mathematics, C.Sankaraiah, V.G.S.Book LinksC.Sankaraiah, V.G.S.Book Links

A text book of Engineering Mathematics, Shahnaz A A text book of Engineering Mathematics, Shahnaz A Bathul, Right PublishersBathul, Right Publishers

A text book of Engineering Mathematics, A text book of Engineering Mathematics, P.Nageshwara Rao, Y.Narasimhulu & N.Prabhakar P.Nageshwara Rao, Y.Narasimhulu & N.Prabhakar Rao, Deepthi PublicationsRao, Deepthi Publications

REFERENCESREFERENCES

A text book of Engineering Mathematics, A text book of Engineering Mathematics, B.V.Raman, Tata Mc Graw HillB.V.Raman, Tata Mc Graw Hill

Advanced Engineering Mathematics, Irvin Advanced Engineering Mathematics, Irvin Kreyszig, Wiley India Pvt. Ltd.Kreyszig, Wiley India Pvt. Ltd.

A text Book of Engineering Mathematics, A text Book of Engineering Mathematics, Thamson Book collectionThamson Book collection

UNIT-VIUNIT-VI

SERIES AND SEQUENCESSERIES AND SEQUENCES

UNIT HEADERUNIT HEADER

Name of the Course: B.TechName of the Course: B.TechCode No:07A1BS02Code No:07A1BS02Year/Branch: I Year Year/Branch: I Year

CSE,IT,ECE,EEE,ME,CIVIL,AEROCSE,IT,ECE,EEE,ME,CIVIL,AEROUnit No: VIUnit No: VI

No. of slides:21No. of slides:21

S. S. No.No.

ModuleModule LectureLectureNo. No.

PPT Slide PPT Slide No.No.

11 Introduction, Comparison Introduction, Comparison test and Auxiliary seriestest and Auxiliary series

L1-5L1-5 8-118-11

22 D’Alembert’s, Cauchy’s, D’Alembert’s, Cauchy’s, Integral, Raabe’s and Integral, Raabe’s and Logarithmic testsLogarithmic tests

L6-10L6-10 12-1612-16

33 Alternating series, Alternating series, Absolute and Conditional Absolute and Conditional convergenceconvergence

L11-13L11-13 17-2117-21

UNIT INDEXUNIT INDEXUNIT-VI UNIT-VI

Lecture-1Lecture-1SEQUENCESEQUENCE

A Sequence of real numbers is a set of A Sequence of real numbers is a set of numbers arranged in a well defined order. numbers arranged in a well defined order. Thus for each positive integer there is Thus for each positive integer there is associated a numbr of the sequence. A associated a numbr of the sequence. A function s:Zfunction s:Z++ → → R is called a SEQUENCE of R is called a SEQUENCE of real numbers.real numbers.

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

CONVERGENT,DIVERGENT, CONVERGENT,DIVERGENT, OSCILLATORY SEQUENCEOSCILLATORY SEQUENCE

If limit of sIf limit of snn=l, then we say that the sequence =l, then we say that the sequence {s{snn} converges to l.} converges to l.

If limit of sIf limit of snn=+=+∞ or -∞ then we say that the ∞ or -∞ then we say that the sequence {ssequence {snn} diverges to l.} diverges to l.

If sequence is neither convergent nor divergent If sequence is neither convergent nor divergent then such sequence is known as an Oscillatory then such sequence is known as an Oscillatory sequence.sequence.

Lecture-2Lecture-2COMPARISON TESTCOMPARISON TEST

If If ΣΣuunn and and ΣΣvvnn are two series of positive terms are two series of positive terms and limit of uand limit of unn/v/vnn = l≠0, then the series = l≠0, then the series ΣΣuunn and and ΣΣvvnn both converge or both diverge. both converge or both diverge.

ExampleExample 1:By comparison test, the series 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 Example 2: By comparison test, the series 2: By comparison test, the series

∑(3n+1)/n(n+2) is divergent∑(3n+1)/n(n+2) is divergent

Lecture-3Lecture-3AUXILIARY SERIESAUXILIARY SERIES

The series The series ΣΣ1/n1/npp converges if p>1 and converges if p>1 and diverges otherwise.diverges otherwise.

Example Example 1: By Auxiliary series test the series 1: By Auxiliary series test the series ∑1/n is divergent since p=1∑1/n is divergent since p=1

ExampleExample 2: By Auxiliary series test the series 2: By Auxiliary series test the series ∑1/n∑1/n3/23/2 is convergent since p=3/2 is convergent since p=3/2>1>1

ExampleExample 3: 3: By Auxiliary series test the series By Auxiliary series test the series ∑1/n∑1/n1/21/2 is divergent since p=1/2 is divergent since p=1/2<1<1

Lecture-4Lecture-4D’ALEMBERT’S RATIO TESTD’ALEMBERT’S RATIO TEST

If If ΣΣuunn is a series of positive terms such that is a series of positive terms such that limit ulimit unn/u/un+1n+1 = l then = l then i) i) ΣΣuun n converges if l>1, converges if l>1, (ii) (ii) ΣΣuunn diverges if l<1, diverges if l<1, (iii) the test fails to decide the nature of (iii) the test fails to decide the nature of the series, if l=1.the series, if l=1.

ExampleExample : By D’Alembert’s ratio test the : By D’Alembert’s ratio test the series ∑1.3.5….(2n-1)/2.4.6…..(2n) xseries ∑1.3.5….(2n-1)/2.4.6…..(2n) xn-1n-1 is is convergent if xconvergent if x>1 and divergent if x<1 or x=1>1 and divergent if x<1 or x=1

Lecture-5Lecture-5CAUCHY’S ROOT TESTCAUCHY’S ROOT TEST

If If ΣΣuunn is a series of positive terms such that limit is a series of positive terms such that limit uunn

1/n1/n =l then =l then (a) (a) ΣΣuunn converges if l<1, converges if l<1, (b) (b) ΣΣuunn diverges if l>1 and diverges if l>1 and (c)the test fails to decide the nature if l=1. (c)the test fails to decide the nature if l=1.

ExampleExample: By Cauchy’s root test the series : By Cauchy’s root test the series ∑[(n+1)/(n+2) x]∑[(n+1)/(n+2) x]nn is convergnt if x is convergnt if x<1 and <1 and divergent if x>1 or x=1.divergent if x>1 or x=1.

Lecture-6Lecture-6INTEGRAL TESTINTEGRAL TEST

Let f be a non-negative decresing function of Let f be a non-negative decresing function of [1,[1,∞). Then the series ∞). Then the series ΣΣuunn and the improper and the improper integral of f(x) between the limits 1 and ∞ integral of f(x) between the limits 1 and ∞ converge or diverge together.converge or diverge together.

ExampleExample 1: By Integral test the series 1: By Integral test the series ∑1/(n∑1/(n22+1) is convergent.+1) is convergent.

ExampleExample 2: By Integral test the series 2: By Integral test the series ∑2n∑2n33/(n/(n44+3) is divergent.+3) is divergent.

Lecture-7Lecture-7RAABE’S TESTRAABE’S TEST

Let Let ΣΣuunn be a series of positive terms and let be a series of positive terms and let limit n[ulimit n[unn/u/un+1n+1 – 1]=l. Then – 1]=l. Then (a) if l>1, (a) if l>1, ΣΣuunn converges converges (b) if l<1, (b) if l<1, ΣΣuunn diverges diverges (c) the test fails when l=1. (c) the test fails when l=1.

ExampleExample: By Raabe’s test the series ∑4.7….: By Raabe’s test the series ∑4.7….(3n+1)/1.2…..n x(3n+1)/1.2…..n xnn is convergent if x is convergent if x<1/3 and <1/3 and divergent if x>1/3 or x=1/3divergent if x>1/3 or x=1/3

Lecture-8Lecture-8LOGARITHMIC TESTLOGARITHMIC TEST

If If ΣΣuunn is a series of positive terms such that is a series of positive terms such that limit n log[ulimit n log[unn/u/un+1n+1]=l, then ]=l, then (a) (a) ΣΣuunn converges if l>1 converges if l>1 (b) (b) ΣΣuunn diverges if l<1 diverges if l<1 (c)the test fails when l=1. (c)the test fails when l=1.

ExampleExample: By logarithmic test the series : By logarithmic test the series 1+x/2+2!/31+x/2+2!/322xx22+….. is convergent if x+….. is convergent if x<e and <e and divergent if x>e or x=edivergent if x>e or x=e

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

TESTTEST Let Let ΣΣuunn be a series of positive terms and let be a series of positive terms and let

limit[{n(ulimit[{n(unn/u/un+1n+1 – 1)-1}logn]=l then – 1)-1}logn]=l then i)i)ΣΣuunn converges for l>1 and converges for l>1 and ii) diverges for l<1.ii) diverges for l<1.

ExampleExample: By Demorgan’s and Bertrand’s test : By Demorgan’s and Bertrand’s test the series 1+2the series 1+222/3/322+2+222/3/322.4.422/5/522+…. is divergent+…. is divergent

Lecture-10Lecture-10ALTERNATING SERIESALTERNATING SERIES

A series whose terms are alternatively positive A series whose terms are alternatively positive and negativ is called an alternating series. An and negativ is called an alternating series. An alternating series may be written as ualternating series may be written as u1 1 – u– u22 + u + u33 -….+(-1)-….+(-1)n-1n-1uunn+……+……

ExampleExample 1:1-1/2+1/3-1/4+….is an alternating 1:1-1/2+1/3-1/4+….is an alternating series.series.

ExampleExample 2: 2:∑(-1)∑(-1)n-1 n-1 n/logn is an alternating n/logn is an alternating seriesseries

Lecture-11Lecture-11LEIBNITZ’S TESTLEIBNITZ’S TEST

If {uIf {unn} is a sequence of positive terms such } is a sequence of positive terms such that that (a)u (a)u11≥u≥u2 2 ≥…. ≥u≥…. ≥un n ≥u≥un+1 n+1 ≥…… ≥…… (b)limit u (b)limit unn=0 then the alternating series is =0 then the alternating series is convergent.convergent.

Example Example 1: By Leibnitz’s test the series 1: By Leibnitz’s test the series ∑(-1)∑(-1)nn/n! is convergent./n! is convergent.

Example Example 2: By Leibnitz’s test the series 2: By Leibnitz’s test the series ∑(-1)∑(-1)nn/(n/(n22+1) is convergent+1) is convergent

Lecture-12Lecture-12ABSOLUTE CONVERGENCEABSOLUTE CONVERGENCE

Consider a series Consider a series ΣΣuunn where u where unn’s are positive or ’s are positive or negative. The series negative. The series ΣΣuunn is said to be is said to be absolutely convergent if absolutely convergent if ΣΣ|u|unn| is convergent. | is convergent.

Example Example 1: The series ∑(-1)1: The series ∑(-1)nn logn/n logn/n22 is is absolute convergence.absolute convergence.

ExampleExample 2: The series 2: The series ∑(-1) ∑(-1)nn (2n+1)/n(n+1)(2n+3) is absolute (2n+1)/n(n+1)(2n+3) is absolute convergence.convergence.

Lecture-13Lecture-13CONDITIONALLY CONVERGENT CONDITIONALLY CONVERGENT

SERIESSERIES If If ΣΣuunn converges and converges and ΣΣ|u|unn| diverges, then we | diverges, then we

say that say that ΣΣuunn converges conditionally or converges conditionally or converges non-absolutely or semi-convergent.converges non-absolutely or semi-convergent.

ExampleExample: The series : The series ∑(-1)∑(-1)nn (2n+3)/(2n+1)(4n+3) is conditional (2n+3)/(2n+1)(4n+3) is conditional convergence.convergence.