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

2. C. F. Gauss
Picture source:

3. Sum for children 8900 + 9003 + 9106 + 9209 + 9312 + 9415 + ···
···
···
10,445 +
10, ··· ,372

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

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

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

7. 2A = (n+1)+(n +1) + … + (n +1) + (n +1)
Arithmetic Series
A ::= … + (n-1) + n
A ::= n + (n-1) + …
2A = (n+1)+(n +1) … + (n +1) + (n +1)
= n(n+1)

8. Arithmetic Series So

9. Geometric Series

10. Geometric Series

11. Geometric Series
xn+1

12. Geometric Series
n+1

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

14. 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.

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

16. 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.

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

18. 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 + 100r r3 + … + 100r10
=100r(1+ r + … + r9)
= 100r(1-r10)/(1-r) = $853.02

19. In-Class Problem
Problems 1 & 2