INTRO TO SEQUENCES AND SERIES VOCABULARY A sequence

INTRO TO SEQUENCES AND SERIES

VOCABULARY �A sequence is a function whose domain is a set of consecutive integers. If a domain is not specified, it is understood that the domain starts with 1. (A list of terms with some pattern) � The values in the range are called the terms of the sequence. � A finite sequence has a limited number of terms. � An infinite sequence continues without stopping. � A sequence can be specified by an equation, or rule.

VOCABULARY CONTINUED � When the terms of a sequence are added together, the resulting expression is a series. � Summation notation or sigma notation, is used to write a series. � For example, in the series 2 i is the index of summation, 1 is the lower limit of summation and 4 is the upper limit of summation.

VOCABULARY CONTINUED �A factorial is the number and each number smaller than it multiplied together. Its notation is ! � For example 5! = 5 * 4 * 3 * 2 * 1 = 120 � On the calculator, factorial is found under math, PRB, option 4 � (-1) raised to an even power = 1 and (-1) raised to an odd power is -1 � For example (-1)6 = 1 and (-1)9 = -1

WRITE TERMS OF SEQUENCES � Write the first four terms of these sequences � 1. 7, 11, 15, 19 � 2. 1/3, 2/9, 6/27, 24/81 � 3. -1, 4, -7, 10

WRITE RULES FOR SEQUENCES � Find the pattern, write a rule for the nth term of the sequence, and then write the next term

Examples 12, 15, 18, 21, 24, 27, 30 Look for a pattern This is going up by 3 each time, this will get multiplied by n, then figure out the 0 th term 12 -3 = 9 an = 3 n + 9 Pattern Next term = 33 0 th term

Examples 34, 27, 20, 13, 6, -1 Look for a pattern This is going down by 7 each time, this will get multiplied by n, then figure out the 0 th term 34+7 = 41 an = -7 n + 41 Pattern Next term = -8 0 th term

Examples -5, -1, 3, 7, 11 Look for a pattern This is going up by 4 each time, this will get multiplied by n, then figure out the 0 th term -5 -4=-9 an = 4 n - 9 Pattern Next term = 15 0 th term

Examples Look for a pattern in the numerator and the denominator separately an =

Examples 1, 4, 9, 16, 25, 36, 49 Term # 1, 2, 3, 4, 5, 6, 7 What are you doing to the term # (domain) in order to get the term (range)? Squaring it! an = n 2

Examples Nothing is happening to numerator so it will not have n in it an =

Sigma notation �Σ (greek letter sigma) means to add Plug in all numbers up until this one Means add It is not important what this letter is Sequence to plug in to 1 st number to plug into ak

Examples � Evaluate 2(3) + 4 36 + 2(4) + 4 + 2(5) + 4

Examples � Evaluate 0 + 1 + 4 + 9 + 16 + 25 + 36 91

Examples � Evaluate 3/6 + 3/7 + 3/8 73/56

WRITE A SERIES USING SUMMATION NOTATION � Figure out the pattern and write formula as before � Figure out which numbers are getting plugged in, use these as the beginning and end numbers in sigma notation

EXAMPLES 1 st number that gets plugged in is 1 (to get 2 in the denominator) last number that gets plugged in is 15 (to get 16 in the denominator)

EXAMPLES 1 st number that gets plugged in is 1 (to get 1 in the denominator) last number that gets plugged in is 20 (to get 400 in the denominator)

Partial Sums Sometimes a series is infinite which means the upper limit is. A partial sum is just the sum of the first few terms of an infinite series. The problem will tell you how many terms to add.

Example � Find the 6 th partial sum of

SUMMATION OF SEQUENCES USING SIGMA NOTATION IN CALCULATOR � List math sum( � List ops seq( � Sum(seq(formula, variable, lower limit, upper limit)) � Examples: � 1. � 2. � 3.

GRAPHING SEQUENCES IN CALCULATOR � Set mode to seq � Put sequence in y= � Press graph � You may need to change the viewing window to see enough point � You can also press table (2 nd graph) to see the (x, y) points

Examples � Graph the first 6 terms