2. Based on Rosen, Discrete Mathematics & Its Applications, 5e Prepared by (c)2001-2004 Michael P. Frank Modified by (c) 2004-2005 Haluk Bingöl 1/18 Module #13 - Summations Bogazici University Department of Computer Engineering CmpE 220 Discrete Mathematics 13. Summations Haluk Bingöl

3. Based on Rosen, Discrete Mathematics & Its Applications, 5e Prepared by (c)2001-2004 Michael P. Frank Modified by (c) 2004-2005 Haluk Bingöl 2/18 Module #13 - Summations Module #13: Summations Rosen 5 th ed., §3.2 ~19 slides, ~1 lecture

4. Based on Rosen, Discrete Mathematics & Its Applications, 5e Prepared by (c)2001-2004 Michael P. Frank Modified by (c) 2004-2005 Haluk Bingöl 3/20 Module #13 - Summations Summation Notation Def. Given a series {a n }, an integer lower bound (or limit) j  0, and an integer upper bound k  j, then the summation of {a n } from j to k is written and defined as follows: Here, i is called the index of summation.

5. Based on Rosen, Discrete Mathematics & Its Applications, 5e Prepared by (c)2001-2004 Michael P. Frank Modified by (c) 2004-2005 Haluk Bingöl 4/20 Module #13 - Summations Generalized Summations Notation. For an infinite series, we may write: To sum a function over all members of a set X={x 1, x 2, …}: Or, if X={x|P(x)}, we may just write:

6. Based on Rosen, Discrete Mathematics & Its Applications, 5e Prepared by (c)2001-2004 Michael P. Frank Modified by (c) 2004-2005 Haluk Bingöl 5/20 Module #13 - Summations Simple Summation Example

7. Based on Rosen, Discrete Mathematics & Its Applications, 5e Prepared by (c)2001-2004 Michael P. Frank Modified by (c) 2004-2005 Haluk Bingöl 6/20 Module #13 - Summations More Summation Examples An infinite series with a finite sum: Using a predicate to define a set of elements to sum over:

8. Based on Rosen, Discrete Mathematics & Its Applications, 5e Prepared by (c)2001-2004 Michael P. Frank Modified by (c) 2004-2005 Haluk Bingöl 7/20 Module #13 - Summations Summation Manipulations Some handy identities for summations: (Distributive law.) (An application of commutativity.) (Index shifting.)

9. Based on Rosen, Discrete Mathematics & Its Applications, 5e Prepared by (c)2001-2004 Michael P. Frank Modified by (c) 2004-2005 Haluk Bingöl 8/20 Module #13 - Summations More Summation Manipulations Other identities that are sometimes useful: (Grouping.) ? (Order reversal.) (Series splitting.)

10. Based on Rosen, Discrete Mathematics & Its Applications, 5e Prepared by (c)2001-2004 Michael P. Frank Modified by (c) 2004-2005 Haluk Bingöl 9/20 Module #13 - Summations 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! And frequently rediscovered by many… Leonhard Euler (1707-1783)

11. Based on Rosen, Discrete Mathematics & Its Applications, 5e Prepared by (c)2001-2004 Michael P. Frank Modified by (c) 2004-2005 Haluk Bingöl 10/20 Module #13 - Summations Euler’s Trick, Illustrated Consider the sum: 1+2+…+(n/2)+((n/2)+1)+…+(n-1)+n We have n/2 pairs of elements, each pair summing to n+1, for a total of (n/2)(n+1). … n+1

12. Based on Rosen, Discrete Mathematics & Its Applications, 5e Prepared by (c)2001-2004 Michael P. Frank Modified by (c) 2004-2005 Haluk Bingöl 11/20 Module #13 - Summations Symbolic Derivation of Trick For case where n is even…

13. Based on Rosen, Discrete Mathematics & Its Applications, 5e Prepared by (c)2001-2004 Michael P. Frank Modified by (c) 2004-2005 Haluk Bingöl 12/20 Module #13 - Summations 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).

14. Based on Rosen, Discrete Mathematics & Its Applications, 5e Prepared by (c)2001-2004 Michael P. Frank Modified by (c) 2004-2005 Haluk Bingöl 13/20 Module #13 - Summations Geometric Progression Def. A geometric progression is a series of the form a, ar, ar 2, ar 3, …, ar k, 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...

15. Based on Rosen, Discrete Mathematics & Its Applications, 5e Prepared by (c)2001-2004 Michael P. Frank Modified by (c) 2004-2005 Haluk Bingöl 14/20 Module #13 - Summations Here we go... Geometric Sum Derivation

16. Based on Rosen, Discrete Mathematics & Its Applications, 5e Prepared by (c)2001-2004 Michael P. Frank Modified by (c) 2004-2005 Haluk Bingöl 15/20 Module #13 - Summations Geometric Sum Derivation...

17. Based on Rosen, Discrete Mathematics & Its Applications, 5e Prepared by (c)2001-2004 Michael P. Frank Modified by (c) 2004-2005 Haluk Bingöl 16/20 Module #13 - Summations Geometric Sum Derivation...

18. Based on Rosen, Discrete Mathematics & Its Applications, 5e Prepared by (c)2001-2004 Michael P. Frank Modified by (c) 2004-2005 Haluk Bingöl 17/20 Module #13 - Summations Nested Summations These have the meaning you’d expect. Note issues of free vs. bound variables, just like in quantified expressions, integrals, etc.

19. Based on Rosen, Discrete Mathematics & Its Applications, 5e Prepared by (c)2001-2004 Michael P. Frank Modified by (c) 2004-2005 Haluk Bingöl 18/20 Module #13 - Summations Some Shortcut Expressions Geometric series. Euler’s trick. Quadratic series. Cubic series.

20. Based on Rosen, Discrete Mathematics & Its Applications, 5e Prepared by (c)2001-2004 Michael P. Frank Modified by (c) 2004-2005 Haluk Bingöl 19/20 Module #13 - Summations Using the Shortcuts Example: Evaluate. Use series splitting. Solve for desired summation. Apply quadratic series rule. Evaluate.

21. Based on Rosen, Discrete Mathematics & Its Applications, 5e Prepared by (c)2001-2004 Michael P. Frank Modified by (c) 2004-2005 Haluk Bingöl 20/20 Module #13 - Summations 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.

22. Based on Rosen, Discrete Mathematics & Its Applications, 5e Prepared by (c)2001-2004 Michael P. Frank Modified by (c) 2004-2005 Haluk Bingöl 21/20 Module #13 - Summations References Rosen Discrete Mathematics and its Applications, 5e Mc GrawHill, 2003Rosen Discrete Mathematics and its Applications, 5e Mc GrawHill, 2003