1 / 11

# Section 9-4 - PowerPoint PPT Presentation

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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Section 9-4' - kobe

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Section 9-4

Sequences and Series

• 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

• 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)

• 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:

• 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:

• many sequences are not arithmetic or geometric

• one famous such sequence is the Fibonacci sequence

• 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

• 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

• arithmetic sequence

• geometric sequence

• 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)

• 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: