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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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