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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 S
- 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 2s 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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