1. Recursive Series Recursion for Series

2. 8/29/2013 Recursive Series 2 The art of asking the right questions in mathematics is more important than the art of solving them − Georg Cantor

3. 8/29/2013 Recursive Series 3 What is a series ? The sum of a sequence of numbers Definitions: A finite series is a sum of form a 1 + a 2 + a 3 + + a n for some positive integer n An infinite series is a sum of form a 1 + a 2 + a 3 + + a n + Series Sequences

4. 8/29/2013 Recursive Series 4 What is a series ? Finite series: a 1 + a 2 + a 3 + + a n Infinite series: a 1 + a 2 + a 3 + + a n + Series Note: A finite series has last term a n An infinite series has no last term

5. 8/29/2013 Recursive Series 5 Example 1 Deposit $1000 at 5% annual interest After five years, with no other deposits or withdrawals, how much is in the account? Let I n = interest accrued in year n Account balance = 1000 + (I 1 + I 2 + I 3 + I 4 + I 5 ) where I 1 =.05(1000) I 2 =.05(1000) +.05I 1 = 1000(.05 +.05 2 ) Series

6. 8/29/2013 Recursive Series 6 n th Partial Sum S n S n = a 1 + a 2 + a 3 +... + a n Finite Series Sum S of all n terms is S n for some n Infinite Series Partial sums form a sequence: S 1 = a 1 S 2 = a 1 + a 2 Partial Sums

7. 8/29/2013 Recursive Series 7 n th Partial Sum S n Partial sums form a sequence: Partial Sums S 1 = a 1 S 2 = a 1 + a 2 S 3 = a 1 + a 2 + a 3 S n = a 1 + a 2 + a 3 + a n What happens to S n as n ∞ ?

8. 8/29/2013 Recursive Series 8 Infinite Series Not immediately clear whether sum S of all terms exists ! Partial sums form a sequence: S 1, S 2, S 3, …, S n,... What happens to S n as Partial Sums ∞ ?

9. 8/29/2013 Recursive Series 9 Infinite Series What happens to S n as n If S n approaches some number S, Otherwise the sum of the series does not exist !! Partial Sums ∞ ? then S is the sumS, written S n of the series

10. 8/29/2013 Recursive Series 10 Examples 1. Sequence is { a n } = { 2 1–n } P Partial sums are Partial Sums S 1 = 1 1 2, …, … = 1,, 1 4, 1 8, 2 n–1 1 S 2 = 1 + 1 2 S 3 = 1 ++ 1 2 1 4 1 8 2 n–1 1 S n = 1 ++ 1 2 1 4 + +

11. 8/29/2013 Recursive Series 11 Examples 1. Sequence is { a n } = { 2 1–n } P Partial Sums 1 2, …, … = 1,, 1 4, 1 8, 2 n–1 1 We can show that 2SnSn ∞ as n Thus S = 2 Graphically 1 1 2 1 4 … 0 1 2

12. 8/29/2013 Recursive Series 12 Examples 2. Partial Sums S 1 = 1 S n = 1 + 2 + 4 + 8 + … + 2 n–1 = 1, 2, 4, 8, 16, … S does not exist ! = a1a1 ( ) 1 – r n 1 – r = –1 + 2 n So … 2. Sequence is Partial sums are { a n } = { 2 n–1 } S 2 = 1 + 2 = 3, S 3 = 1 + 2 + 4 = 7, We can show that ∞ SnSn ∞ as n

13. 8/29/2013 Recursive Series 13 Finite Series Sum of n terms can be written as Details Summation Notation ∑ akak k=1 n General Term Index Initial Index Value Final Index Value S = a 1 + a 2 + a 3 +... + a n = k=1 n akak ∑

14. 8/29/2013 Recursive Series 14 Finite Series Examples Summation Notation = 2(1) + 2(2) + 2(3) + 2(4) + 2(5) = 30 = 6 + 6 = 24 ∑ 2k k=1 5 A. ∑ 6 k=1 4 B. = 2 ∑ k k=1 5

15. 8/29/2013 Recursive Series 15 Example Sequence: Partial Sums n th partial sums are: { an }{ an } = { 2n – 1 (–1) n–1 4 }, 3 – 4– 4 = 1 4,, …, 5 4 3 – 4– 4 SnSn k=1 n 2k – 1 (–1) k–1 4 ∑ = … 2n – 1 (–1) n–1 4 + 3 –4 = 1 4 + ++ + 5 4 7

16. 8/29/2013 Recursive Series 16 Example Partial Sums Does S n approach a value as n =  ≈ 3.14159265358979323846... Question: ∞ ? as n ∞  We can show that S n Thus 2k – 1 (–1) k–1 4 = ∑= ∑ k=1 ∞ S SnSn n 2k – 1 (–1) k–1 4 ∑ =

17. 8/29/2013 Recursive Series 17 Notation Manipulation Consider sequence { a n } and constant c Factoring: Example Summation Notation ∑ k=1 n akak c = ca1 + ca2 + ca3 + … + canca1 + ca2 + ca3 + … + can = c( a 1 + a 2 + a 3 + … + a n ) ∑ k=1 n akak c = ∑ 2k k=1 5 = ∑ k 5 2 = 2(1 + 2 + 3 + 4 + 5) = 30

18. 8/29/2013 Recursive Series 18 Notation Manipulation Consider sequences { a n }, { b n } Addition: Example Summation Notation k=1 5 ∑ k 2+ ( ) = 5(5 + 1) 2 + 5(2)5(2) = 25 = ∑ k=1 n akak + ∑ n bkbk ∑ n (ak +bk)(ak +bk) = ∑ 5 k + ∑ 5 2 Special Sum

19. 8/29/2013 Recursive Series 19 Think about it !