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Module #13: Summations

Module #13: Summations. 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.

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Module #13: Summations

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  1. Module #13:Summations

  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.

  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:

  4. Simple Summation Example

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

  6. Summation Manipulations • Some handy identities for summations: (Distributive law.) (Applicationof commut-ativity.) (Index shifting.)

  7. More Summation Manipulations • Other identities that are sometimes useful: (Series splitting.) (Grouping.)

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

  9. Some Shortcut Expressions(1) Geometric series. Euler’s trick. Quadratic series. Cubic series.

  10. Some Shortcut Expressions(2)

  11. Using the Shortcuts • Example: Evaluate . • Use series splitting. • Solve for desiredsummation. • Apply quadraticseries rule. • Evaluate.

  12. Example

  13. 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.

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