Discrete Mathematics and its Applications 5/3/2019 University of Florida Dept. of Computer & Information Science & Engineering COT 3100 Applications of Discrete Structures Dr. Michael P. Frank Slides for a Course Based on the Text Discrete Mathematics & Its Applications (5th Edition) by Kenneth H. Rosen A word about organization: Since different courses have different lengths of lecture periods, and different instructors go at different paces, rather than dividing the material up into fixed-length lectures, we will divide it up into “modules” which correspond to major topic areas and will generally take 1-3 lectures to cover. Within modules, we have smaller “topics”. Within topics are individual slides. 5/3/2019 (c)2001-2003, Michael P. Frank (c)2001-2002, Michael P. Frank

Rosen 5th ed., §3.2 ~19 slides, ~1 lecture Module #12: Summations Rosen 5th ed., §3.2 ~19 slides, ~1 lecture 5/3/2019 (c)2001-2003, Michael P. Frank

Summation Notation Given a series {an}, an integer lower bound (or limit) j0, and an integer upper bound kj, then the summation of {an} from j to k is written and defined as follows: Here, i is called the index of summation. 5/3/2019 (c)2001-2003, Michael P. Frank

Generalized Summations For an infinite series, we may write: To sum a function over all members of a set X={x1, x2, …}: Or, if X={x|P(x)}, we may just write: 5/3/2019 (c)2001-2003, Michael P. Frank

Simple Summation Example 5/3/2019 (c)2001-2003, Michael P. Frank

More Summation Examples An infinite series with a finite sum: Using a predicate to define a set of elements to sum over: 5/3/2019 (c)2001-2003, Michael P. Frank

Summation Manipulations Some handy identities for summations: (Distributive law.) (Application of commut- ativity.) (Index shifting.) 5/3/2019 (c)2001-2003, Michael P. Frank

More Summation Manipulations Other identities that are sometimes useful: (Series splitting.) (Order reversal.) (Grouping.) 5/3/2019 (c)2001-2003, Michael P. Frank

Example: Impress Your Friends Boast, “I’m so smart; give me any 2-digit number n, and I’ll add all the numbers from 1 to n in my head in just a few seconds.” I.e., Evaluate the summation: There is a simple closed-form formula for the result, discovered by Euler at age 12! Leonhard Euler (1707-1783) 5/3/2019 (c)2001-2003, Michael P. Frank

Euler’s Trick, Illustrated Consider the sum: 1+2+…+(n/2)+((n/2)+1)+…+(n-1)+n n/2 pairs of elements, each pair summing to n+1, for a total of (n/2)(n+1). n+1 … n+1 n+1 5/3/2019 (c)2001-2003, Michael P. Frank

Symbolic Derivation of Trick 5/3/2019 (c)2001-2003, Michael P. Frank

Concluding Euler’s Derivation So, you only have to do 1 easy multiplication in your head, then cut in half. Also works for odd n (prove this at home). 5/3/2019 (c)2001-2003, Michael P. Frank

Example: Geometric Progression A geometric progression is a series of the form a, ar, ar2, ar3, …, ark, where a,rR. The sum of such a series is given by: We can reduce this to closed form via clever manipulation of summations... 5/3/2019 (c)2001-2003, Michael P. Frank

Geometric Sum Derivation Here we go... 5/3/2019 (c)2001-2003, Michael P. Frank

Derivation example cont... 5/3/2019 (c)2001-2003, Michael P. Frank

Concluding long derivation... 5/3/2019 (c)2001-2003, Michael P. Frank

Nested Summations These have the meaning you’d expect. Note issues of free vs. bound variables, just like in quantified expressions, integrals, etc. 5/3/2019 (c)2001-2003, Michael P. Frank

Some Shortcut Expressions Geometric series. Euler’s trick. Quadratic series. Cubic series. 5/3/2019 (c)2001-2003, Michael P. Frank

Using the Shortcuts Example: Evaluate . Use series splitting. Solve for desired summation. Apply quadratic series rule. Evaluate. 5/3/2019 (c)2001-2003, Michael P. Frank

Summations: Conclusion You need to know: How to read, write & evaluate summation expressions like: Summation manipulation laws we covered. Shortcut closed-form formulas, & how to use them. 5/3/2019 (c)2001-2003, Michael P. Frank