1. SEQUENCES A function whose domain is the set of all integers greater than or equal to some integer n 0 is called a sequence. Usually the initial number n 0 is taken to be 0 or 1. The typical value of the function is usually denoted by a n rather than f(n) or f(x). a n is referred to as the nth term of the sequence. The sequence whose nth term is a n is sometimes also denoted by {a n } or.

2. Limit of a Sequence Formal Definition: The sequence {a n } converges to the limit L, if given any  > 0, there exists a corresponding integer N such that |a n  L| N. Notation: lim a n = L or a n  L n   The limit of a sequence is unique If no such L exists, the sequence is said to diverge

3. Subsequences If the terms of a sequence appear in another sequence in their given order, the first sequence is called a subsequence of the second. If a sequence {a n } converges to L, so do all its subsequences. If any subsequence of a sequence {a n } diverges, or two subsequences converge to different limits, then {a n } diverges

4. Basic Properties of Sequences - 1 Proposition 11: a) The limit of a convergent sequence is unique. b) A convergent sequence is bounded. c) An unbounded sequence is divergent. d) A convergent (divergent) sequence remains convergent (divergent) if any or all of its first k terms are altered. e) If a sequence {a n } converges to L, so do all its subsequences. f) If any subsequence of a sequence {a n } diverges, or two subsequences converge to different limits, then {a n } diverges.

5. Results about Limits of Sequences - 1 Proposition 12: Suppose the sequences {a n } and {b n } converge to L and M respectively. Then: Sum/Difference Rule: {a n  b n }  L  M Product Rule: {a n. b n }  L. M Constant Multiple Rule: {ca n }  cL, where c is constant Quotient Rule: {a n / b n }  L / M, provided M  0, and b n  0 for all n

6. Results about Limits of Sequences - 3 Proposition 13 (Continuous Function Theorem for Sequences): Suppose the sequence {a n } converges to L. If f is a function that is continuous at L and defined for all a n, then {f(a n )} converges to f(L). Proposition 14: Suppose that f(x) is a function defined for all x  n 0 and that {a n } is a sequence such that a n = f(n) for n  n 0. Then: Lim f(x) = L implies a n  L x 

7. Fundamental Result Proposition 15 (Completeness Property of Real Numbers): Every set X of real numbers which is bounded above has a least upper bound. Remark: a proof of the above requires a deep understanding of the real number system R and is beyond the scope of this course. Remark: However, note that the above does not apply to the system of rational numbers Q. Roughly speaking, the implication of the above is that there are gaps in the rational numbers but not in the real numbers. That is why it is called the completeness property.

8. Basic Result about Sequences Definition: A sequence {a n } is said to be non- decreasing if a n  a n+1 for all n. Proposition 16: A non-decreasing sequence of real numbers converges if and only if it is bounded above. If so, it converges to its lub. Remark 1: The above is a consequence of the Completeness Property of Real Numbers. Proof is left as an exercise. Remark 2: Similar definitions can be given for non- increasing sequences, and a parallel result holds for them also.

9. Cauchy Sequences Definition: A sequence {a n } is said to be a Cauchy sequence if given any  > 0, there exist a positive integer N such that |a n  a m | N. Remark: Roughly speaking, a sequence is Cauchy if its terms get closer and closer to each other. Proposition 17: A sequence of real numbers is convergent if and only if it is a Cauchy sequence. Remark: The above is again a consequence of the Completeness Property of Real Numbers. However, the proof is difficult and is not within the scope of the course.

10. Table of Frequent Limits NB: all the limits below are taken as n  1. Lim ln n /n = 0 (Proof: Apply Prop. 14) 2. Lim n 1/n = 1 (Proof: Take logs and apply the above) 3. Lim x 1/n = 1 for any fixed x > 0 (Proof: As for 2.) 4. Lim x n = 0 for any fixed x, |x| < 1 5. Lim (1 + x/n) n = e x for any fixed x 6. Lim x n = 0 for any fixed x n!

11. INFINITE SERIES Given a sequence of numbers {a n }, we can construct the sequence of partial sums: s 1 = a 1 s 2 = a 1 + a 2 s 3 = a 1 + a 2 + a 3 n s n = a 1 + a 2 + a 3 + … + a n =  a k k = 1 This sequence of partial sums is called an infinite series, with nth term a n.

12. Limit of an Infinite Series If the sequence of partial sums converges to a limit L, we say the infinite series converges to the sum L. Notation:  a 1 + a 2 + a 3 + … + a n + … =  a n = L k = 1 If the sequence of partial sums of the series does not converge, the series is said to diverge. However, we generally refer to an infinite series by the notation  a n, in both the cases (convergent or divergent).

13. Basic Test for Divergence Proposition 18: If  a n converges, then a n  0 Proof: left as an exercise Remark: As a consequence of the above, we have: Corresponding Test for Divergence: If a n does not converge to 0, or if lim a n fails to exist, then the series must diverge Caution: A series can diverge even in the case a n  0 (this is not a sufficient condition for convergence) Famous Counter-Example: the harmonic series  1/n diverges even though 1/n  0

14. Combining Series Proposition 19: If  a n = A and  b n = B are two convergent series, then: a) Sum Rule:  (a n + b n ) =  a n +  b n = A + B b) Difference Rule:  (a n  b n ) =  a n   b n = A  B c) Constant Multiple Rule:  ca n = c  a n = cA (any constant Adding and Deleting Terms: A finite number of terms can be added to or deleted from an infinite series without affecting its convergence/divergence. However, the sum of a convergent series would change if some initial terms are added/deleted.

15. Direct Comparison Test Proposition 20: Let  a n be a series of non-negative terms. Then:  a n converges if there is a convergent series  c n with a n  c n for all n > N  a n diverges if there is a divergent series of non- negative terms  d n with a n  d n for all n > N (Here N is some fixed integer depending on the series under consideration)

16. For Applying Comparison Tests Some useful convergent series: Geometric series with  r  < 1 The series  1/n! The p-series  1/n p with p >1 Some useful divergent series: Geometric series with  r  > 1 The harmonic series  1/n The p-series  1/n p with p <1

17. Comparison Tests  Example 1:The series  1 + cos n n=1 n 2 Compare with the convergent series  2/n 2 Since 1 + cos n  2/n 2 for n  1, n 2 the series is convergent by Direct Comparison Test

18. Comparison Tests  Example 2:The series  ln(n + 1)/(n + 1) n=1 Compare with the divergent series  1/(n + 1) Let a n = ln(n + 1)/(n + 1), and b n = 1/(n + 1) Since lim a n / b n = lim ln(n + 1) =  n   n   the series is divergent by Limit Test

19. Limit Comparison Test Proposition 21: Suppose a n > 0 and b n > 0 for all n  N (an integer). Then: If lim (a n / b n ) = c > 0, then  a n and  b n n   both converge or both diverge If lim (a n / b n ) = 0 and  b n converges, n   then  a n converges If lim (a n / b n ) =  and  b n diverges, n   then  a n diverges

20. Intrinsic Tests - 1 Proposition 22 (Ratio Test): Let  a n be a series with positive terms, and suppose that : lim (a n+1 / a n ) = . Then: n   the series converges if  < 1 the series diverges if  > 1 or  is infinite the test is inconclusive if  = 1

21. Intrinsic Tests - 2 Proposition 23 (n-th Root Test): Let  a n be a series with positive terms, and suppose that : lim ( n  a n ) = . Then: n   the series converges if  < 1 the series diverges if  > 1 or  is infinite the test is inconclusive if  = 1 Proof: left as an exercise (similar to Ratio Test)

22. Examples for Intrinsic Tests - 1  Example 1:The series  n 10 /10 n n=1 Then a n +1 /a n = (n + 1) 10 /10 n+1  n 10 /10 n = (1/10) (1 + 1/n) 10  (1/10) < 1, the series converges by Ratio Test

23. Examples for Intrinsic Tests - 2  Example 2:The series  (log n) n /n n n=1 Converges by applying root test: n  a n = log n/n  0 < 1

24. Alternating Series A series in which terms are alternately positive and negative is called an alternating series. Proposition 24 (Leibniz’s Theorem for Alternating Series): The series   (  1) n+1 u n = u 1  u 2 + u 3  u 4 + … n = 1 converges if all three of the following conditions are satisfied: All u n are positive u n  u n+1 for all n  N ( N some fixed integer) u n  0

25. Absolute and Conditional Convergence - 1 A series  a n is said to be absolutely convergent if the corresponding series of absolute values   a n  converges. A series which converges but does not converge absolutely is said to be conditionally convergent.

26. Absolute and Conditional Convergence - 2 Proposition 25 (Absolute Convergence Test): If the series:    a n  converges, then so does the series  a n n = 1 n = 1 Proposition 26 (Re-Arrangement Theorem): If the series  a n converges absolutely, and b 1, b 2,b 3, …, b n,.. is any re-arrangement of the sequence {a n }, then  b n also converges absolutely, and   b n =  a n n = 1 n = 1 Remark: The proof of Proposition 26 is difficult, you may refer to an advanced calculus (or real analysis) book.

27. Convergence for Complex Sequences and Series Remark: Complex sequences and series do not require extensive separate treatment because of the following result: Proposition 27: Suppose that z n = x n + iy n and L = X + iY. Then: a) the sequence converges to L if and only if converges to X and converges to Y b) the series  z n converges to L if and only if  x n converges to X and  y n converges to Y. Remark: The following results for real series also hold for complex series: a) A necessary (but not sufficient) condition for the convergence of the series  z n is that z n  0. b) Absolute convergence of a complex series implies convergence.