The document outlines the contents of a mathematics textbook covering topics like ordinary differential equations, linear differential equations, mean value theorems, functions of several variables, and vector calculus. It lists recommended textbooks and references for further study. It provides an overview of the unit on series and sequences, outlining 13 lectures covering topics such as comparison tests, auxiliary series, ratio tests, root tests, alternating series tests, and absolute and conditional convergence.

1. MATHEMATICS-I

2. 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

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,
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

4. 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

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

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

9. 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.

10. 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

11. 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

12. 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

13. 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.

14. 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 ∑2n 3/
(n4+3) is divergent.

15. 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

16. 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>1
(b) Σun diverges if l<1
(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

17. 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

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 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

19. 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

20. 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.

21. 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.