Appendix E: Sigma Notation

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Appendix E Sigma Notation
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Definition Sequence

• A sequence is a function a(n) (written an) whos
  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, an1, , where an is a
  term in a sequence, an-1 is the term before it,
  and an1 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 whos input for the next term in the
  sequence is the value of the previous term is
  called a recursive sequence.

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Definition Arithmetic Sequence
An arithmetic sequence is 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. Example Find the general term
of the 7/3, 8/3, 3, 10/3, .
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Definition Geometric Sequence

• A geometric sequence is a sequence generated by
  multiplying the previous term by a real number
  (called the common ratio r). 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.
• Example Find the general term sequence 2, 2/5,
  2/25, 2/125,
• TI seq(ax , x, i start, i stop)

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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.

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

• For a finite arithmetic series,
• For an infinite arithmetic series,
• For a finite geometric series,
• For an infinite geometric series,
  if r lt 1. It DNE otherwise.

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