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

natural number

n

to each

a n

represents the value of the n th term

Sequences

• a sequence can be defined “explicitly” using a formula to find

a n ex

.

a n

 3

n

 4 • a sequence can be defined “recursively” by a formula relating each term to its

ex

previous term(s) .

a n



a n

 1  6 and

a

1 14, 8, 2,  4,  10,  14

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:

a n a

1 (

n

1)

d

• the recursive formula for arithmetic seq. is:

a n



a n

 1 

d

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:

a n a r

1

n

 1 • the recursive formula for geometric seq. is:

a n

 (

a n

 1 ) 

r

Fibonacci Sequence

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

a a n

1  1 

a n

 2 

a

2

a n

  1 1 1, 1, 2, 3, 5, 8, 13, 21,

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,

a k

, = 1 to n is denoted: from

k n k

  1

a k k

is called the index

Partial Sums

• the sum of the first n terms of a sequence is called “the n th partial sum” • the symbol S n sum” is used for the “nth partial • 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 S n

Partial Sum Formulas

• arithmetic sequence

S n



n

2 (

a

1 

a n

) 

n

2 ( 2

a

1  (

n

 1 )

d

) • geometric sequence

S n



a

1  1  1 

r r n

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: if

r

 1 , then the series

k

   1

a

1 

r n

 1 converges to 1

a

1 

r