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Appendix E: Sigma Notation

Appendix E: Sigma Notation. Definition: Sequence. A sequence is a function a ( n ) (written a n ) who’s domain is the set of natural numbers {1, 2, 3, 4, 5, ….}. a n is called the general term of the sequence.

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Appendix E: Sigma Notation

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  1. Appendix E: Sigma Notation

  2. Definition: Sequence • A sequenceis a function a(n) (written an) who’s domain is the set of natural numbers {1, 2, 3, 4, 5, ….}. an is called the general term of the sequence. • The output of a sequence can be written as {a1, a2, a3, …, an-1, an, an+1, …}, where an is a termin a sequence, an-1 is the term before it, and an+1 is the term after it. • Sequences can be either finite(their domains are {1, 2, 3, …, n}) or infinite(their domains are {1, 2, 3, ….}). • A sequence who’s input for the next term in the sequence is the value of the previous term is called a recursive sequence.

  3. Definition: Arithmetic Sequence An arithmetic sequenceis a sequence generated by adding a real number (called the common difference,d) to the previous term to get the next term. The general term of an arithmetic is given by an = a1 + d(n – 1) where a1 and d are any real numbers. ExampleFind the general term of the 7/3, 8/3, 3, 10/3, ….

  4. Definition: Geometric Sequence A geometric sequenceis a sequence generated by multiplying the previous term by a real number (called the common ratior). The general term of a geometric sequence is given by an = a1 r(n – 1) where a1 and r are any real numbers, is called an geometric sequence. ExampleFind the general term sequence 2, 2/5, 2/25, 2/125, … TI: seq(ax , x, i start, i stop)

  5. Definition: Series • A finite series is the sum of a finite number of terms of a sequence. • An infinite series is the sum of an infinite number of terms of a sequence. • We use sigma notation to denote a series. The series does not have to start at i = 1, but i must be in the domain of ai.

  6. Definition: Geometric Sequence The nth partial sum is the sum of the first n terms of a sequence. It MUST start at i = 1 with partial sum notation. An infinite sum is the sum of all the terms of an infinite sequence.

  7. Definition: Example TI: sum(seq(ax , x, i start, i stop))

  8. Definition: Example

  9. Definition: Series • For a finite arithmetic series, • For an infinite arithmetic series, • For a finite geometric series, • For an infinite geometric series, if | r | < 1. It DNE otherwise.

  10. Definition: Example

  11. Definition: Series Formulas Let c be a constant and n a positive integer.

  12. Definition: Series Formulas 9. Write a formula for the series in terms of n: 10. If the interval [a, b] is split into n equal subintervals, write a sequence xi that represents the x coordinate of the left side, midpoint, and right side of each subinterval. 11. Show that

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