Chapter 10

1. Chapter 10 Sequences and Series

2. 10.1 Sequences and Summation Notation Definition:A sequence is a function f whose domain is the set of natural numbers. The values are called terms of the sequence.

3. Example: nth term formula Sequence First Term Second Term Fourth Term nth Term Third Term

4. Recursive Sequences Definition:A sequence that is defined by listing the first term, or the first few terms, and then describes how to determine the remaining terms from the given ones is recursive. Example:

5. Partial Sums of a Sequence For the sequence the partial sums are

6. Sigma Notation Given we can write the sum of the first n terms using summation notation, or sigma notation. k is called the index of summation, or summation variable.

7. Example End with 5 Start with 1

8. Properties of Sums

9. 10.2 Arithmetic Sequences Definition:An arithmetic sequence is a sequence of the form The number a is the first term, and d is the common difference of the sequence. The nth term of an arithmetic sequence is given by

10. Partial Sums of Arithmetic Sequences For the arithmetic sequence, The nth partial sum Is given by either of the following formulas. 1. 2.

11. 10.3 Geometric Sequences Definition: A geometric sequence is a sequence of the form The number a is the first term, and r is the common ratio of the sequence. The nth term of an arithmetic sequence is given by

12. Partial Sums of Geometric Sequences For the geometric sequence the nth partial sum is given by

13. Sum of an Infinite Geometric Series A sum of the form is called an infinite series. If then the infinite series has the sum

14. 10.5 Mathematical Induction Consider the following sums. Look for a pattern. The sum of the first 1 odd number is The sum of the first 2 odd numbers is The sum of the first 3 odd number is The sum of the first 4 odd number is The sum of the first 5 odd number is

15. Can we make a conjecture about the sum of the first n odd integers? The sum of the first n odd integers is What is the form of any odd integer? Mathematically, our conjecture now reads …

16. Can we prove it? A proof is a clear argument that demonstrates the truth of a statement beyond doubt. To prove our conjecture, we will use a special kind of proof called mathematical induction.

17. Mathematical Induction To prove something using induction, we need to establish a sequence of mathematical statements. We call these statements P1, P2, P3, etc. The sum of the first 1 odd number is The sum of the first 2 odd numbers is The sum of the first 3 odd number is

18. The Key Idea Suppose we can prove that whenever one of these statements is true, the statement following it is also true. For every k, if is true, then is true. This is called the induction step.

19. Principle of Mathematical Induction For each natural number n, let P(n) be a statement depending on n. Suppose the following two conditions are satisfied. • P(1) is true. • For every natural number k, if P(k) is true, then P(k+1) is true. Then P(n) is true for all natural number n.

20. How It Works. • Show P(1) is true. The induction step then leads through if P(1) is true, then P(2) is true. If P(2) is true, then P(3) is true, etc. • Assume P(k) is true. • Use P(k) (and some algebra) to show P(k+1) is true.

21. Prove: For all natural numbers n, Step 1: Show P(1) true. Step 2: Assume P(k) is true. Step 3: Show P(k+1) is true.

22. Manipulate the left hand side so it looks like the right hand side.

23. We have now shown that if P(k) is true, P(k+1) is also true. The induction step is completed. Hence, by the Principle of Mathematical Induction, for all natural numbers n,