1. Mathematics Medicine 2010-2011 Sequences and series

2. Sequences Definition A sequence is an ordered set of numbers Each number in the sequence is called a term of the sequence. The term corresponding to the positive integer n is the n-th term of the sequence and is denoted by a symbol such as bn. The sequence whose n-th term is bn is denoted {bn}. If sequences have a finite number of terms, they are called finite sequences. Some sequences go on for ever, and these are called infinite sequences.

3. Sequences When the second procedure is used to define a sequence, the sequence is said to be defined recursively or by means of a recursion formula. There are two common ways used to describe a sequence: 1. Stating the general term. For example: 2. Giving the first term, b 1, and stating the relationship between each term and its successor. For example:

4. Arithmetic sequence Definition An arithmetic sequence is a sequence with general term: where a 1 is the first term and d is a constant (common difference).

5. Geometric sequence Definition A geometric sequence is a sequence with general term: where a 1 is the first term and r is a constant (common ratio).

6. Convergent and divergent sequence. If a sequence has a limit, it is a convergent sequence. If a sequence has no limit, it is a divergent sequence. The notation is read: “The sequence {an} converges to A” or “The limit of {an} is A”. It can happen that as we move along the sequence the terms get closer and closer to a fixed value A (called limit).

7. The Limit of a Sequence EXAMPLE 1. The terms of sequence 1 are very small for large values of n. The terms are clustered very close to zero for large n, but zero is not a term of the sequence. Zero is the limit of the sequence. The sequence is said to converge to zero. EXAMPLE 2. Each term in sequence 2 is three greater than its predecessor. For large values of n, the terms are large. There is no number about which the terms of the sequence cluster. This sequence has no limit; it is said to diverge. EXAMPLE 3. Sequence 3 is similar to sequence 1. The terms are close to zero for large values of n, but unlike sequence 1, there are terms to the right and to the left of zero. This sequence converges to zero. EXAMPLE 4. The terms of sequence 4 are very close to 3 for large values of n. Since the terms are close to 3, the sequence has a limit of 3. EXAMPLE 5. The terms of sequence 5 which correspond to even numbers are close to 2. The terms corresponding to odd numbers are close to -2. Consequently there is not a unique single number about which the terms cluster. The sequence has no limit. Therefore it is to diverge.

8. The discussion of the examples suggests the following ideas: a. Some sequences have limits; some do not. b. If a sequence has a limit, then the limit is a unique number. c. If a sequence has a limit, then most of the terms are close to the limit.

9. Series and Their Sums Definition A series is an indicated sum of the terms of a sequence. EXAMPLE 1. Given the arithmetic sequence a, a + d, a + 2d,…, a + (n-1)d the related arithmetic series isa + (a + d) + (a + 2d) + … + [a + (n -1) d EXAMPLE 2. Given the geometric sequence a, ar, ar 2, ar 3, …, ar n-1 the related geometric series isa + ar + ar 2 + ar 3 +… +ar n-1

10. Partial sum Definition Given a series. The nth partial sum, Sn, is defined by the equation Definition Given an infinite series The sum of the infinite series is defined to be the limit of the sequence of partial sums {Sn} where If {Sn} has a limit, the series is a convergent series. If {Sn} has no limit, the series is a divergent series. for all i; in the set of natural numbers.

11. Arithmetic series If the terms of an arithmetic sequence are added, the result is known as an arithmetic series. The sum of the first n terms of an arithmetic series with first term a and common difference d is denoted by S n and given by

12. Geometrics series If the terms of a geometric sequence are added, the result is known as a geometric series. The sum of the first n terms of a geometric series with first term a and common ratio r is denoted by S n and given by provided r is not equal to 1 The formula excludes the use of r = 1 because in this case the denominator becomes zero, and division by zero is never allowed

13. Sum of an infinite sequence When the terms of an infinite sequence are added we obtain an infinite series. But in in some cases sum is finite and can be found. Consider the special case of an infinite geometric series for which the common ratio r lies between -1 and 1. In such a case the sum always exists, and its value can be found from the following formula: The sum of an infinite number of terms of a geometric series is denoted by S ∞ and is given by provided -1 <r< 1 Note that if the common ratio is bigger than 1 or less than -1, that is r>1 or r<-1, then the sum of an infinite geometric series cannot be found.

14. Limits of Special Sequences Theorem Any sequence with a > 0 and a  R converges to zero For example Theorem Any sequence {c} of constants c converges to c. For example: {5}  5 Theorem If | r | < 1, then {rn} converges to 0. For example:

15. Operations with Sequences Definitions Given any two sequences {an} and {bn}, then the sum of {an} and {bn} is the sequence with n th term an + bn; –that is {an}+{bn}={an+bn}. the difference of {an} and {bn} is the sequence with n th term an - bn; –that is {an}-{bn}={an-bn}. the product of {an} and {bn} is the sequence with n th term an  bn; –that is {an}  {bn}={an  bn}. the quotient of {an} and {bn} is the sequence with n th term that is {an}  {bn}=

16. Theorem If {an}  A and {bn}  B, then a.{an+bn}  A+B. b. {an-bn}  A-B. c. {an  bn}  A  B. d. (B  0)