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Module #12: Summations

Module #12: Summations. Rosen 5 th ed., §3.2 ~19 slides, ~1 lecture. 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:

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Module #12: Summations

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

  2. 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. (c)2001-2003, Michael P. Frank

  3. 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: (c)2001-2003, Michael P. Frank

  4. Simple Summation Example (c)2001-2003, Michael P. Frank

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

  6. Summation Manipulations • Some handy identities for summations: (Distributive law.) (Applicationof commut-ativity.) (Index shifting.) (c)2001-2003, Michael P. Frank

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

  8. 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! LeonhardEuler(1707-1783) (c)2001-2003, Michael P. Frank

  9. 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 (c)2001-2003, Michael P. Frank

  10. Symbolic Derivation of Trick (c)2001-2003, Michael P. Frank

  11. 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). (c)2001-2003, Michael P. Frank

  12. 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... (c)2001-2003, Michael P. Frank

  13. Geometric Sum Derivation • Herewego... (c)2001-2003, Michael P. Frank

  14. Derivation example cont... (c)2001-2003, Michael P. Frank

  15. Concluding long derivation... (c)2001-2003, Michael P. Frank

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

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

  18. Using the Shortcuts • Example: Evaluate . • Use series splitting. • Solve for desiredsummation. • Apply quadraticseries rule. • Evaluate. (c)2001-2003, Michael P. Frank

  19. 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. (c)2001-2003, Michael P. Frank

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