2. Module #12: Summations Rosen 5 th ed., §3.2 Based on our knowledge on sequence, we can move on to summations easily.

3. Summation Notation 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. Here is the notation for summation and sigma is used again. Note that sigma in the string is the set of symbols but now it means the sum of elements

4. Generalized Summations 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:

5. Simple Summation Example This slides shows a simple example of summation. The lower bound is 2 and upper bound is 4. and the generating function is the square of the index plus 1. What we need to do is just to add elements or the results of the generating functions as the index runs from 2 to 4.

6. More Summation Examples An infinite series with a finite sum: Using a predicate to define a set of elements to sum over:

7. Summation Manipulations Some handy identities for summations: Index shifting * Two kinds of shifting: lower, higher

8. More… Order reversal Splitting grouping

9. Example: Impress Your Friends “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!

10. Terminology: closed-form An equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally accepted set. For example, an infinite sum would generally not be considered closed-form.

11. 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 Suppose n is even (n = 2k)

12. Symbolic Derivation of Trick Suppose n is even (n = 2k)

13. 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

14. Example: Geometric Progression 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... * Arithmetic progression

15. Geometric Sum Derivation

16. Derivation example cont...

17. Concluding long derivation...

18. Nested Summations These have the meaning you’d expect. Note issues of free vs. bound variables, just like in quantified expressions, integrals, etc.

19. Some Shortcut Expressions Geometric series. Euler’s trick. Quadratic series. Cubic series.

20. Using the Shortcuts Example: Evaluate. –Use series splitting. –Solve for desired summation. –Apply quadratic series rule. –Evaluate.

21. 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.