Download - Number Sequences Lecture 5 (chapter 4.1 of the book and chapter 9 of the notes) ? overhang
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Number Sequences
Lecture 5
(chapter 4.1 of the book and chapter 9 of the notes)
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Examples
a1, a2, a3, …, an, …
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.
1st + 30th = 89 + 466 = 5552nd + 29th = (1st+13) + (30th13) = 5553rd + 28th = (2nd+13) + (29th13) = 555
So the sum is equal to 15x555 = 8325.
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Arithmetic Series
Given n numbers, a1, a2, …, an with common difference d, i.e. ai+1 - ai =d.
What is a simple closed form expression of the sum?
Adding the equations together gives:
Rearranging and remembering that an = a1 + (n − 1)d, we get:
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Geometric Series
2 n-1 nnG 1+x +x + +x::= +x
What is the closed form expression of Gn?
2 n-1 nnG 1+x +x + +x::= +x
2 3 n n+1nxG x +x +x + +x +x=
GnxGn= 1 xn+1
n+1
n
1- xG =
1- x
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Infinite Geometric Series
n+1
n
1- xG =
1- x
Consider infinite sum (series)
2 n-1 n i
i=0
1+x+x + +x + =x + x
n+1n
nn
1-lim x 1limG
1- x 1-=
x=
for |x| < 1 i
i=0
1x =
1- x
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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
bX 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/b2 today.
The Future Value of Money
$n paid k years from now
is only worth $n/bk 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 + 100r2 + 100r3 + + 100r10
= 100r(1+ r + + r9)
= 100r(1r10)/(1r) = $853.02
$n paid k years from now
is only worth $n/bk today
Annuities
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Annuities
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 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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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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1x+1
0 1 2 3 4 5 6 7 8
1
1213
12
1 13
Harmonic Number
Estimate Hn:
n
1 1 1H ::=1+ + + +
2 3 n
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n
0
1 1 1 1 dx 1 + + +...+
x+1 2 3 n
n+1
n1
1dx H
x
nln(n+1) H
Now Hn as n , so
Harmonic series can go to infinity!
Integral Method (OPTIONAL)
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How far out?
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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)n
i=1
ln(i)
Factorial
How to estimate n!?
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…ln 2ln 3ln 4
ln 5ln n-1
ln nln 2
ln 3ln 4ln 5
ln n
2 31 4 5 n–2 n–1 n
ln (x+1)ln (x)
Integral Method (OPTIONAL)
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ln(x) dx ln(i) ln (x+1)dxi=1
nn n
1 0
x
lnxdx =xlne
Reminder:
n
i=1
1 nln(i) n+ ln
2 eso guess:
n ln(n/e) +1 ln(i) (n+1) ln((n+1)/e) +1
Analysis (OPTIONAL)
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exponentiating:
nn
n! n/ e e
n
i=1
1 nln(i) n+ ln
2 e
nn
n! 2πne
~Stirling’s formula:
Stirling’s Formula