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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 an to each natural number n
- an represents the value of the nth term

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

- 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

- 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

- many sequences are not arithmetic or geometric
- one famous such sequence is the Fibonacci sequence

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

- 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

- arithmetic sequence
- geometric sequence

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

- 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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