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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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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

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Summation

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A Telescoping Sum When do we have closed form formulas?

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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.

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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:

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

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

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Some Examples

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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.

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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

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$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

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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

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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

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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.

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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

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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

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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.

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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.

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

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

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

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

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How far out? ? overhang Book Stacking

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

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Product

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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!?

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… 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)

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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)

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exponentiating: Stirling’s formula: Stirling’s Formula

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