M1 unit vi-jntuworld

532 views

Published on

Published in: Technology, Education
0 Comments
1 Like
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total views
532
On SlideShare
0
From Embeds
0
Number of Embeds
3
Actions
Shares
0
Downloads
15
Comments
0
Likes
1
Embeds 0
No embeds

No notes for slide

M1 unit vi-jntuworld

  1. 1. MATHEMATICS-I
  2. 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. 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. 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. 5. UNIT-VISERIES AND SEQUENCES
  6. 6. UNIT HEADER Name of the Course: B.Tech Code No:07A1BS02 Year/Branch: I YearCSE,IT,ECE,EEE,ME,CIVIL,AERO Unit No: VI No. of slides:21
  7. 7. UNIT INDEX UNIT-VIS. Module Lecture PPT SlideNo. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.

×