Mathematics for Computer Science MIT 6.042J/18.062J Sums, Products & Asymptotics Copyright © Albert Meyer, 2002. Prof. Albert Meyer & Dr. Radhika Nagpal

C. F. Gauss Picture source: http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Gauss.html

Sum for children 8900 + 9003 + 9106 + 9209 + 9312 + 9415 + ··· 8900 + 9003 + 9106 + 9209 + 9312 + 9415 + ··· 9930 + ··· 10,445 + 10,960 + ··· +11,372

Sum for children Nine-year old Gauss (so the story goes) saw that each number was 103 greater than the previous one.

Sum for children 8900 + (8900+103) + (8900+2·103) + (8900+3·103) + ··· + (8900+24·103) = 8900·25 + (1 + 2 + 3 + ··· + 24)103

A ::= 1 + 2 + … + (n-1) + n A = 1 + 2 + … + (n-1) + n Arithmetic Series A ::= 1 + 2 + … + (n-1) + n A = 1 + 2 + … + (n-1) + n

2A = (n+1)+(n +1) + … + (n +1) + (n +1) Arithmetic Series A ::= 1 + 2 + … + (n-1) + n A ::= n + (n-1) + … + 2 + 1 2A = (n+1)+(n +1) + … + (n +1) + (n +1) = n(n+1)

Arithmetic Series So

Geometric Series

Geometric Series

Geometric Series 1 - xn+1

Geometric Series n+1

The future value of $$. Annuities I will promise to pay you $100 in exactly one year, if you will pay me $X now.

1.03 X = 100. Annuities My bank will pay me 3% interest. If I deposit your $X for a year, I can’t lose if 1.03 X = 100.

Annuities I can’t lose if you pay me: X = $100/1.03 ≈ $97.09

Annuities 97.09¢ today is worth $1.00 in a year $1.00 in a year is worth $1/1.03 today $n in a year is worth $nr today, where r = 1/1.03.

Annuities $n in two years is worth $nr2 today $n in k years is worth $nr k today

Annuities I will pay you $100/year for 10 years If you will pay me $Y now. I can’t lose if you pay me 100r + 100r2 + 100r3 + … + 100r10 =100r(1+ r + … + r9) = 100r(1-r10)/(1-r) = $853.02

In-Class Problem Problems 1 & 2