Sums & 1 + x + x 2 + + x n G = n x - x 2 - - x n - x n + 1 - PowerPoint PPT Presentation

Mathematics for Computer Science Geometric Series MIT 6.042J/18.062J Sums & 1 + x + x 2 + ⋯ + x n G = n x - x 2 - ⋯ - x n - x n + 1 Money − xG = − n Albert R Meyer, April 10, 2013 geometric-sum.1 Albert R Meyer, April 10, 2013 geometric-sum.2 Geometric Series Geometric Series 1 + x + x 2 + ⋯ + x n 1 + x + x 2 + ⋯ + x n G = G = n n x - x 2 - ⋯ - x n - x n + 1 x - x 2 - ⋯ - x n - x n + 1 − xG = − − xG = − n n Albert R Meyer, April 10, 2013 geometric-sum.3 Albert R Meyer, April 10, 2013 geometric-sum.4 1

Geometric Series Geometric Series 1 + x + x 2 + ⋯ + x n G = G − xG = n n n x - x 2 - ⋯ - x n - x n + 1 − xG = − n 1 - x n+1 1 - x n+1 Albert R Meyer, April 10, 2013 geometric-sum.5 Albert R Meyer, April 10, 2013 geometric-sum.6 Geometric Series Geometric Series n+1 1-x G = n 1-x G − xG = 1 - x n+1 n n Consider infinite sum (series) 1 − x n + 1 G = ∞ 1+ x + x 2 + ⋯ + x n-1 + x n + ⋯ = ∑ x i n 1 − x i=0 Albert R Meyer, April 10, 2013 geometric-sum.7 Albert R Meyer, April 10, 2013 geometric-sum.8 2

Infinite Geometric Series Infinite Geometric Series ∞ 1-x n+1 1 G n = ∑ i x = 1-x 1-x i=0 n+1 1-lim x 1 →∞ limG n = n = for |x| < 1 1-x 1-x n →∞ Albert R Meyer, April 10, 2013 geometric-sum.9 Albert R Meyer, April 10, 2013 geometric-sum.10 The future value of $$ The future value of $$ My bank will pay me 3% interest. define bankrate I will pay you $100 in 1 year, b ::= 1.03 if you will pay me $X now. � bank increases my $$ by this factor in 1 year. Albert R Meyer, April 10, 2013 geometric-sum.11 Albert R Meyer, April 10, 2013 geometric-sum.12 3

The future value of $$ The future value of $$ If I deposit your $X now, • $1 in 1 year is worth $0.9709 now. I will have $b � � X in 1 year. • $r last year is worth $1 today, So I won � t lose money as long as where r ::= 1/b. b � X ≥ 100 • So $n paid in 2 years is worth X ≥ $100/1.03 ≈ $97.09 $nr paid in 1 year, and is worth $nr 2 today. Albert R Meyer, April 10, 2013 geometric-sum.13 Albert R Meyer, April 10, 2013 geometric-sum.14 The future value of $$ Annuities I pay you $100/year for 10 years, if you will pay me $Y now. $n paid k years from now I can � t lose if you pay me is worth $n·r k today 100r + 100r 2 + 100r 3 + � + 100r 10 where r ::= 1/bankrate. = 100r(1+ r + � + r 9 ) = 100r(1 � r 10 ) / (1 � r) = $853.02 Albert R Meyer, April 10, 2013 geometric-sum.15 Albert R Meyer, April 10, 2013 geometric-sum.16 4

Annuities Manipulating Sums I pay you $100/year for 10 years,  n+1   n  d d 1-x ∑ i x = if you will pay me $853.02.     dx dx 1-x     QUICKIE: If bankrates unexpectedly i=0 increase in the next few years,  n+1  n n 1 d 1-x ∑ ∑ A. You come out ahead i-1 i ix = ix =   x dx 1-x   B. The deal stays fair i=0 i=1 C. I come out ahead Albert R Meyer, April 10, 2013 geometric-sum.17 Albert R Meyer, April 10, 2013 geometric-sum.18 Manipulating Sums n+1 n+2 n x-(n+1)x +nx ∑ i-1 ix = 2 (1-x) i=1 Albert R Meyer, April 10, 2013 geometric-sum.19 5

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