Section 9-4

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

An Image/Link below is provided (as is) to download presentation

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

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: