Block 3 Discrete Systems Lesson 10 –Sequences and Series Both finite and countable infinite series and much more one two three four five six seven eight nine ten 1 Narrator: Charles Ebeling University of Dayton

Summation Notation 2

Defining Sequences S n is the n th term in a sequence that may be finite or infinite S n is a function defined on the set of natural numbers Examples: If S n = 1/n, then the first 4 terms in the sequence are 1, 1/2, 1/3, 1/4 If S n = 1/n!, then the first 4 terms in the sequence are 1, 1/2, 1/6, 1/24 The general term for the sequence -1, 4, -9, 16, -25 is S n = (-1) n n 2 3

Arithmetic Progression S n = a + (n-1)(d) is an arithmetic progression starting at a and incrementing by d For example, the first six terms of S n = 3 + (n-1) (4) are 3, 7, 11, 15, 19, 23 4

The limit of an infinite sequence If for an infinite sequence, s 1, s 2, …, s n, … there exists an arbitrarily small  > 0 and an m > 0 such that |s n – s| m, then s is the limit of the sequence. 5

Examples 6

Series The sum of a sequence is called a series. The sum of an infinite sequence is called an infinite series If the series has a finite sum, then the series is said to converge; otherwise it diverges A finite sum will always converge Let S n = s 1 + s 2 + … + s n S n is the sequence of partial sums 7

Convergence Necessary condition for convergence – Sufficient condition for divergence - harmonic series: 1 + 1/2 + 1/3 + … + 1/n + … diverges and 8

The p-series converges for p > 1 and divergence for p  1 9

The Arithmetic Series The sum of the first 100 odd numbers is 10

More to do with arithmetic series 11 For the overachieving student: Prove these results by induction

The Geometric Sequence 12

The Geometric Series S n = a +ar + ar 2 + … + ar n-1 r S n = ar + ar 2 + … + ar n-1 + ar n S n - r S n = (1-r) S n = a - ar n This is a most important series. 13

The Geometric Series in Action Find the sum of the following series: 14

15 Future Value of an Annuity The are n annual payments of R (dollars) where the annual interest rate is r. Let S = the future sum after n payments, then S = R + R(1+r) + R(1+r) 2 + … + R(1+r) n-1 012…n-2n-1n 0RR…RRR0RR…RRR R(1+r) n-1 R(1+r) R(1+r) 2

16 More Future Value of an Annuity the sum of a finite geometric series

The Binomial Theorem A really Big Bonus. Isaac Newton’s first great discovery (1676) 17

First, a notational diversion… Factorial notation: n! = 1·2·3 ··· (n-2) ·(n-1) ·n where 0! = 1! = 1 and n! = n (n-1)! for example: A useful fact: 18

Here it is…For n integer: 19

Some Observations on the binomial theorem n+1 terms sum of the exponents in each term is n coefficients equi-distant from ends are equal A related theorem: This is truly an amazing result! Look for my triangle on the next slide. 20

Pascal’s triangle 21

The Generalized Binomial Theorem For n non-integer or negative: 22

Other series worth knowing binomial series exponential series logarithmic series 23

This Series has come to an end Next Time – Discrete Probability 24