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: