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Section 9-4

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

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  1. Section 9-4 Sequences and Series

  2. 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

  3. 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)

  4. 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:

  5. 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:

  6. Fibonacci Sequence • many sequences are not arithmetic or geometric • one famous such sequence is the Fibonacci sequence

  7. 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

  8. 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

  9. Partial Sum Formulas • arithmetic sequence • geometric sequence

  10. 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)

  11. 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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