Appendix A: Summations Motivation: Evaluating and/or bounding sums are frequently needed in the solution of recurrences Two types of evaluation problems: Prove by induction that formula is correct Find the function that the sum equals or is bounded by Encountered both types in analysis of insertion sort

Prove by induction that  j=1 to n j = n(n+1)/2 Called the arithmetic sum Text p 1059

Use the arithmetic sum to evaluate the sums in the analysis of insertion sort runtime

Important sums to remember Arithmetic  k=1 to n k = n(n+1)/2 =  (n 2 ) Geometric  k=0 to n x k = (x n+1 – 1)/(x – 1) when x  1 Harmonic  k=1 to n (1/k) = ln(n) +  (1)

Alternate forms of geometric sum useful in tree analysis  k=0 to n-1 x k = (x n – 1)/(x – 1) when x  1 How do we show this is true?  k=0 to ∞ x k = 1/(1 – x) when |x| < 1

Integration and differentiation can be used to evaluate sums derivative: d{  f(x)}/dx =  df/dx integral:  dx {  f(x)} =   dx f(x) Example: eq. A.8 p1148 Show  k=0 to ∞ k x k = x/(1 – x) 2 when 0< |x| < 1

See bottom p1147 for simpler approach

Bounding sums Prove a bound by induction Bound ever term in sum Bound by integration monotone increasing and decreasing summands

Prove by induction on integers that  k=0 to n 3 k = O(3 n )

there exist c=4/3 such that 0<4<3c Similar argument applies n=2, etc. Property of sums independent of what we are trying to prove and (1/3 +1/c) 3/2; therefore, c=3/2 or larger will work in the definition of big O Hence  k=0 to n 3 k = O(3 n ) by definition 3 Base case n=0 is true < c3 n which implies

Example of bound sum by bounding every term Show that (n/2) 2 <  k=1 to n k < n 2

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

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

Use bounding by integrals for informal proof that  k=1 to n k -1 =  (ln(n))

CptS 450 Spring 2015 [All problems are from Cormen et al, 3nd Edition] Homework Assignment 3: due 2/4/15 1.ex A.1-3 p ex A.1-6 p ex A.2-1 p 1156 (hint: use integration) 4.part a of prop A-1 p 1156 using bound each term