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Number Sequences Lecture 5 (chapter 4.1 of the book and chapter 9 of the notes) ? overhang.

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Presentation on theme: "Number Sequences Lecture 5 (chapter 4.1 of the book and chapter 9 of the notes) ? overhang."— Presentation transcript:

1 Number Sequences Lecture 5 (chapter 4.1 of the book and chapter 9 of the notes) ? overhang

2 Examples a 1, a 2, a 3, …, a n, … 1,2,3,4,5,6,7,… 1/2, 2/3, 3/4, 4/5,… 1,-1,1,-1,1,-1,… 1,-1/4,1/9,-1/16,1/25,… General formula

3 Summation

4 A Telescoping Sum When do we have closed form formulas?

5 Sum for Children 89 + 102 + 115 + 128 + 141 + 154 + ··· + 193 + ··· + 232 + · · · + 323 + ··· + 414 + ··· + 453 + 466 Nine-year old Gauss saw 30 numbers, each 13 greater than the previous one. 1 st + 30 th = 89 + 466 = 555 2 nd + 29 th = (1 st +13) + (30 th  13) = 555 3 rd + 28 th = (2 nd +13) + (29 th  13) = 555 So the sum is equal to 15x555 = 8325.

6 Arithmetic Series Given n numbers, a 1, a 2, …, a n with common difference d, i.e. a i+1 - a i =d. What is a simple closed form expression of the sum? Adding the equations together gives: Rearranging and remembering that a n = a 1 + (n − 1)d, we get:

7 Geometric Series What is the closed form expression of G n ? G n  xG n =1  x n+1

8 Infinite Geometric Series Consider infinite sum (series) for |x| < 1

9 Some Examples

10 The Value of an Annuity Would you prefer a million dollars today or $50,000 a year for the rest of your life? An annuity is a financial instrument that pays out a fixed amount of money at the beginning of every year for some specified number of years. Examples: lottery payouts, student loans, home mortgages. A key question is what an annuity is worth. In order to answer such questions, we need to know what a dollar paid out in the future is worth today.

11 My bank will pay me 3% interest. define bankrate b ::= 1.03 -- bank increases my $ by this factor in 1 year. The Future Value of Money So if I have $X today, One year later I will have $bX Therefore, to have $1 after one year, It is enough to have b  X  1. X  $1/1.03 ≈ $0.9709

12 $1 in 1 year is worth $0.9709 now. $1/b last year is worth $1 today, So $n paid in 2 years is worth $n/b paid in 1 year, and is worth $n/b 2 today. The Future Value of Money $n paid k years from now is only worth $n/b k today

13 Someone pays you $100/year for 10 years. Let r ::= 1/bankrate = 1/1.03 In terms of current value, this is worth: 100r + 100r 2 + 100r 3 +  + 100r 10 = 100r(1+ r +  + r 9 ) = 100r(1  r 10 )/(1  r) = $853.02 $n paid k years from now is only worth $n/b k today Annuities

14 I pay you $100/year for 10 years, if you will pay me $853.02. QUICKIE: If bankrates unexpectedly increase in the next few years, A.You come out ahead B.The deal stays fair C.I come out ahead

15 Loan Suppose you were about to enter college today and a college loan officer offered you the following deal: $25,000 at the start of each year for four years to pay for your college tuition and an option of choosing one of the following repayment plans: Plan A: Wait four years, then repay $20,000 at the start of each year for the next ten years. Plan B: Wait five years, then repay $30,000 at the start of each year for the next five years. Assume interest rate 7% Let r = 1/1.07.

16 Plan A: Wait four years, then repay $20,000 at the start of each year for the next ten years. Plan A Current value for plan A = 114,666.69

17 Plan A: Wait four years, then repay $20,000 at the start of each year for the next ten years. Plan A Current value for plan A

18 Plan B Plan B: Wait five years, then repay $30,000 at the start of each year for the next five years. Current value for plan B = 93,840.63.

19 Plan B Current value for plan B Plan B: Wait five years, then repay $30,000 at the start of each year for the next five years.

20 Profit $25,000 at the start of each year for four years to pay for your college tuition. Loan office profit = $3233.

21 Profit $25,000 at the start of each year for four years to pay for your college tuition. Loan office profit = $3233.

22 1 x+1 0 1 2 3 4 5 6 7 8 1 1212 1313 1212 1 1313 Harmonic Number Estimate H n :

23 Now H n   as n  , so Harmonic series can go to infinity! Integral Method (OPTIONAL)

24 How far out? ? overhang Book Stacking

25 The classical solution Harmonic Stacks Using n blocks we can get an overhang of

26 Product

27 Factorial defines a product: Turn product into a sum taking logs: ln(n!) = ln(1·2·3 ··· (n – 1)·n) = ln 1 + ln 2 + ··· + ln(n – 1) + ln(n) Factorial How to estimate n!?

28 … ln 2 ln 3 ln 4 ln 5 ln n-1 ln n ln 2 ln 3 ln 4 ln 5 ln n 23145n–2n–1n ln (x+1) ln (x) Integral Method (OPTIONAL)

29  ln(x) dx   ln(i)   ln (x+1)dx i=1 n n n 1 0 Reminder: so guess: n ln(n/e) +1   ln(i)  (n+1) ln((n+1)/e) +1 Analysis (OPTIONAL)

30 exponentiating: Stirling’s formula: Stirling’s Formula


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