1. Quiz 1, Monday 1/28/19
Chapters 1,2 and 3
HW assignments 1-5

2. Appendix A: Summations Sk=1 to n f(k)
Two types of summation problems in algorithms:
For a given lower limit, find the function of the upper limit
that the sum is equal to.
a) prove by induction on integers
b) transform a sum we know into the one we need
2) Find an asymptotic bound for very large values of the
upper limit

3. Example: induction on integers
prove
Setup to use “if S(n-1) then S(n)”
inductive hypothesis
application

4. Change of summation index
Example from analysis of insertion sort:
use to evaluate
Let k=j-1 then

5. Integration and differentiation can be used to
evaluate a sum using one you know
derivative: d{ f(x)}/dx =  df/dx
integral: dx {f(x)} =  dx f(x)
Example: use geometric series
To show k=0 to ∞ k xk = x/(1 – x)2
when 0< |x| < 1

6. Homework Assignment 6: due 1/25/19
ex A.1-3 p 1149
use derviatives on the geometric series to show that

7. Bounding sums
bound sum by bounding every term
Show that (n/2)2 < Sk=1 to n k < n2

9. Bound by integration: monotone increasing summand
Shaded area is integral of
continuous function f(x)
Sum equals area of “upper sum”
rectangles
Same f(x) different limits
on integration
Sum equals area of “lower sum”
rectangles

10. Note the difference for monotone increasing and decreasing summand
Method not applicable if summand is not monotone increasing or decreasing

11. Use bounding by integrals for informal proof that
Sk=1 to n k-1 = Q(ln(n)) called the “harmonic” sum

13. Homework Assignment 7: due 1/30/19
1) Use integration to show that is bounded from above by a constant .
2)Find asymptotic upper and lower bounds on
by bounding every term.
Make your lower bound tighter than w(n).

14. Important sums to remember
Arithmetic k=1 to n k = n(n+1)/2 = Q(n2)
Geometric k=0 to n xk = (xn+1 – 1)/(x – 1)
when x  1
Harmonic k=1 to n (1/k) = ln(n) + O(1)
Geometric series k=0 to ∞ xk = 1/(1 – x)
when |x| < 1