2. (1) An ordered set of real numbers is called a sequence and is denoted by ( ). If the number of terms is unlimited, then the sequence is said to be an infinite sequence and is its general term. For instance (i) 1,3,5,7,…,(2n-1),…, 1. sequences 1

3. (ii) 1,1/2,1/3,…,1/n,..., (iii) 1,-1,1,-1,…, (-1),… are infinite sequences. (2) Limit. A sequence is said to tend to a limit, if for every ε >0, a value N of n can be found such that for n ≥ N. We then write or simply as 2

4. (3) Convergence. If a sequence ( ) has a finite limit, it is called a convergent sequence. If ( ) is not convergent, it is said to be divergent. In the above examples,(ii) is convergent, while (i) and (iii) are divergent. 3

5. (4) Bounded sequence. A sequence ( ) is said to be bounded,if there exists a number such that for every n. (5) Monotonic sequence. the sequence ( ) is said to increase steadily or to decrease steadily according as or for all values of. 4

6. Both increasing and decreasing sequences are called monotonic sequences. A monotonic sequence always tends to alimit, finite or infinite. Thus, a sequence which is monotonic and bounded is convergent. 5

7. (1) Def.if be an infinite sequence of real numbers, then Is called an infinite series. An infinite series is denoted by and the sum of its first terms is denoted by. 2. series 6

8. (2) Convergence, divergence and oscillation of a series. Consider the infinite series And let the sum of the first terms be Clearly, is a function of and as increases indefinitely three possibilities arise. 7

9. (i)If tends to a finite limit as, the series is said to be convergent. (ii) If tends to as,the series is said to be divergent. (iii) If does not tend to a unique limit as, then the series is said to be oscillatory or non-convergent. 8

10. Examine for convergence the series.. Here Hence this series is divergent. Test the convergence of the series.. Here terms 9

11. =0,5 or 1 according as the number of terms is 3m,3m+1, 3m+2. clearly in this case, does not tend to a unique limit. Hence the series is oscillatory. Geometric series Show that the series (i)Converges if,(ii) diverges if, and (iii) oscillates if. 10

12. . Let Case. When. Also so that The series is convergent. 11

13. Case. (i) When r >1,. Also so that. The series is divergent. (ii) When r =1,then and the series is divergent. 12

14. Case. (i) When r =-1, then the series becomes 1-1+1- 1+1-1….which is an oscillatory series.. (ii) When r 1 Then and as. according as n is even or odd. Hence the series oscillates. 13

15. The truth of the following properties is self –evident and these may be regarded as axioms : 1.The convergence or divergence of an infinite series remains unaffected by the addition or removal of a finite number of its terms ; for the sum of these terms being 3. GENERAL PROPERITIES OF SERIES 14

16. the finite quantity does not on addition or removal alter the nature of its sum. 2. If a series in which all the terms are positive is convergent, the series remains convergent even when some or all of its terms are negative ; for the sum is clearly the greatest when all the terms are positive. 15

17. 3. The convergence or divergence of an infinite series remains unaffected by multiplying each term by a finite number. 4. Series of positive terms 1.An infinite series in which all the terms after some particular term are positive term series. 16

18. e.g., -7-5-2+2+7+13+20+……is a positive term series as all its terms after the third are positive. 2. A series of positive terms either converges or diverges to + ; for the sum of its first n terms, omitting the negative terms, tends to either a finite limit or +. 17

19. 3. Necessary condition for convergence. If a positive term series is convergent, then Let since is given to be convergent. finite quantity k (say). Also But Hence the result. 18

20. Obs. 1. It is important to note that the converse of this result is not true. Consider, for instance, the series Since the term go on descending, i.e. 19

21. Thus the series is divergent even though Hence is a necessary but not sufficient condition for convergence of. Obs. 2. The above result leads to a simple test for divergence : If, the series must be divergent. 20

22. If two positive term series and be such that (i) converges, (ii) for all values of n, then also converges. Proof. Since is convergent, 5. COMPARISON TESTS 21

23. a finite quantity k (say) Also since Adding, Hence the series also converges. 22

24. Obs. If, however,the relation holds for values of n greater than a fixed number m, then the first m terms of both the series can be ignored without affecting their convergence or divergence. II. If two positive term series and be such that : 23

25. (i) diverges, (ii) for all values of n, then also diverges. Its proof is similar to that of Test I. III. Limit form If two positive term series and be such that finite quantity ( ), then and converge or diverge together. 24

26. Proof. Since, a finite number ( ) By definition of a limit, there exists a positive number, however small, such that for or for 25

27. Omitting the first m terms of both the series, we have for all n ….. (1) Case I When is convergent, then a finite number …..(2) Also from (1), i.e. for all n. by (2) 26

28. Hence is also convergent. Case II. When is divergent, then ….(3) Also from (1) or for all n [by (3) Hence is also divergent. 27

29. A positive term series f(1)+f(2)+……+f(n)+….., where f(n) decreases as n increases, converges or diverges according as the integral is finite or infinite. 6. INTEGRAL TEST 28

30. The area under the curve y=f(x), between any two ordinates lies between the set of inscribed and escribed rectangles formed by ordinates at x=1,2,3,…… as in Fig 6.1 Then 29

31. Fig 6.1 Taking limits as,we find from the second inequality that limit 30

32. Hence if integral (1) is finite, so is limit. Similarly, from the first inequality, we see that if the integral (1) is infinite, so is limit.But the given series either converges or diverges to +, i.e. limit either finite or infinite Hence the result follows. 31

33. show that the harmonic series of order p i.e. Converges for p>1 and diverges for p 1. By the above test, this series will converge or diverge according as is finite or infinite. 32

34. If p 1,, i.e. finite for p>1 for p<1 If p=1,,this proves the result. 33

35. . test for convergence the series. We have Take Then =2, which is finite and non –zero. 34

36. Both and converge or diverge together. But is known to be convergent. Hence is also convergent. Test the convergence of the series : (i) (ii) 35

37. . (i) We have (Expanding by Binomial Theorem) Taking, We have which is finit and non zero. 36

38. Both and converge or diverge together. But is known to be divergent. Hence is also divergent. (ii) When x<1, comparing the given series with, We get [ 37

39. But is convergent, so is also convergent. When x>1, comparing with,we get But is convergent, so is also convergent. When x=1, Which is divergent. Hence, converges for x 1 but diverges for x=1. 38

40. If and be two positive term series, then converges if (i) converges,and (ii) from and after some particular term, 7. COMPARISON OF RATIOS 8. D’ALEMBERT’S RATIO TEST In a positive term series if, then the series converges for and diverges for. 39

41. Test for convergence the series (i) (ii). (i)We have and 40

42. =.Hence converges if and diverges if. If,then, Taking,we get, afinite quantity. Both and converge or diverge together. But is a convergent series. 41

43. is also convergent. Hence the given series converges if and diverges if. (ii) Here Thus by Ratio test, converges for x 1. But it fail for x=1. 42

44. When x=1, diverges for x=1. Hence the given series converges for x<1 and diverges for x 1. Discuss the convergence of the series (i) (ii) 43

45. . (i) We have Hence the given series is convergent. (ii) Given series is Hence the given series is convergent. 44

46. (1) Def. A series in which the terms are alternately positive or negative is called an alternating series. (2) Leibnitz’s rule*. An alternating series converges if (i) each term is numerically less than its preceding term, and (ii) 9. ALTERNATING SERIES 45

47. If,the given series is oscillatory. The given series is Suppose …(1) and …(2) Consider the sum of 2n terms. It can be written as …(3) or as …(4) 46

48. By virtue of (1), the expressions within the brackets in (3) and (4) are all positive. It follows from (3) that is positive and increases with n. Also from (4), We note that always remains less than. Hence must tend to a finite limit. 47

49. Moreover [by (2) Thus tends to the same finite limit whether n is even or odd. Hence the given series is convergent. When The given series is oscillatory. 48

50. Discuss the convergence of the series (i) (ii) (i) The terms of the given series are alternately positive and negative ; each term is numerically less than its preceding term 49

51. Also Hence by Leibnitz’s rule,the given series is convergent. (ii) The terms of the given series are alternately positive and negative and i.e. Also Hence by Leibnitz’s rule, the given series is oscillatory. 50

52. Examine the character of the series (i) (ii) (i) The terms of the given series are alternately positive and negative ; and each term is numerically less than its preceding term. 51

53. But which is not zero. Hence the given series is oscillatory. (ii) The terms of the given series are alternately positive and negative i.e., Also Hence the given series is convergent. 52

54. The series of positive terms and the alternating series are special types of these series with arbitrary signs. Def. (1) If the series of arbitrary terms be such that the series is convergent, then the series is said to be absolutely convergent. 10.SERIES OF POSITIVE OR NEGATIVE TERMS 10.SERIES OF POSITIVE OR NEGATIVE TERMS 53

55. (2) If is divergent but is convergent, then is said to be conditionally convergent. For instance, the series is absolutely convergent, since the series is known to be convergent. 54

56. Again,since the alternating series is convergent, and the series of absolute values is divergent, so the original series is conditionally convergent. 55

57. Test whether the series is convergent or not ? The series of absolute terms is which is, evidently convergent. The given series is absolutely convergent and hence it is convergent. 56

58. Test for convergence and absolute convergence. The terms of the given series are alternately positive or negative. Also each term is numerically less than the preceding term and 57

59. Hence, the given series is convergent by Leibnitz’s rule. Also which is a finite quantity. Thus the given series is absolutely convergent. Test the series for conditional convergence 58

60. Here Then i.e. Also Thus by Leibnitz’s rule, and therefore is convergent. 59

61. Also. Taking we note that Since is divergent, therefore is also divergent. i.e. Is convergent but is divergent. Thus the given series is conditionally convergent. 60