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Section 9-4. Sequences and Series. Sequences. a sequence is an ordered progression of numbers they can be finite (a countable # of terms) or infinite (continue endlessly) a sequence can be thought of as a function that assigns a unique number a n to each natural number n

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section 9 4

Section 9-4

Sequences and Series

sequences
Sequences
  • a sequence is an ordered progression of numbers
  • they can be finite (a countable # of terms) or infinite (continue endlessly)
  • a sequence can be thought of as a function that assigns a unique number an to each natural number n
  • an represents the value of the nth term
sequences1
Sequences
  • a sequence can be defined “explicitly” using a formula to find an
  • a sequence can be defined “recursively” by a formula relating each term to its previous term(s)
arithmetic sequences
Arithmetic Sequences
  • an arithmetic sequence is a special type of sequence in which successive terms have a common difference (adding or subtracting the same number each time)
  • the common difference is denoted d
  • the explicit formula for arithmetic seq. is:
  • the recursive formula for arithmetic seq. is:
geometric sequences
Geometric Sequences
  • a geometric sequence is a special type of sequence in which successive terms have a common ratio (multiplying or dividing by the same number each time)
  • the common ratio is denoted r
  • the explicit formula for geometric seq. is:
  • the recursive formula for geometric seq. is:
fibonacci sequence
Fibonacci Sequence
  • many sequences are not arithmetic or geometric
  • one famous such sequence is the Fibonacci sequence
summation notation
Summation Notation
  • summation notation is used to write the sum of an indefinite number of terms of a sequence
  • it uses the Greek letter sigma: Σ
  • the sum of the terms of a sequence, ak, from k = 1 to n is denoted:

k is called the index

partial sums
Partial Sums
  • the sum of the first n terms of a sequence is called “the nth partial sum”
  • the symbol Sn is used for the “nth partial sum”
  • some partial sums can be computed by listing the terms and simply adding them up
  • for arithmetic and geometric sequences we have formulas to find Sn
partial sum formulas
Partial Sum Formulas
  • arithmetic sequence
  • geometric sequence
infinite series
Infinite Series
  • when an infinite number of terms are added together the expression is called an “infinite series”
  • an infinite series is not a true sum (if you add an infinite number of 2’s together the sum is not a real number)
  • yet interestingly, sometimes the sequence of partial sums approaches a finite limit, S
  • if this is the case, we say the series converges to S (otherwise it diverges)
infinite geometric series
Infinite Geometric Series
  • there are several types of series that converge but most are beyond the scope of this course (Calculus)
  • one type that we do study is an infinite geometric series with a certain property:
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