Transcription of Lecture Notes - Sequences
1 SequencesLecture Notes for Section is an infinite list of numbers written in a definite order:sequence# % ) "' $# The numbers in the list are called the of the sequence. In the sequence above, the firsttermsterm is , the second term is , the third term is , and so forth, with each successive term being#%)twice the previous term. How can we figure out the 10th term in this sequence? Well, we could simply continuedoubling until we arrive at the tenth term:# % ) "' $# '% "#) #&' &"# "!#%However, a better method would be to find a for the sequence, a formula for howformulathe th term depends on . In this case, each term is a power of :88## # # % # ) # "' # $# "#$%&In particular, the formula for the th term of this sequence is.
2 Thus the 10th term must be ,8##8"!which is ."!#%Different Ways of Writing a Sequence It's often clearer when writing a sequence to provide a formula for the th term method is to include the formula among the list of terms:# % ) "' # 8 Sometimes, it is convenient to write the formula for a sequence. The convention is that anyonlyformula surrounded by braces specifies a sequence:e fe f##888 "_or simply. There is also a convention for discussing Sequences abstractly. When talking about asequence in general, we will write the terms using variables:+ + + + "#$%To avoid running out of different letters, we use the same letter for all the variables (in thiscase ), with subscripts to distinguish between different terms.
3 Such a sequence may also be+written using braces:e fe f++888 "_or for Sequences The trick to finding the formula for a sequence is to recognize the pattern, and figure out how todescribe it in terms of . Here are a few simple examples:8 EXAMPLE 1 Find formulas for the following Sequences : (a)(b)(c)" % & ' ( ) " % * "' #& " " " "# $ % &SOLUTION(a) This is the sequence .e f" 8(b) Usually a good way of figuring out the formula is to make a table showing and :8 +88 " # $ % & + % & ' ( ) 8As you can see, is always three greater than , so this is the sequence .+88 $8ef(c) We make a table showing and :8 +88 " # $ % & + " % * "' #& 8As you can see, the th term is equal to , so this is the sequence.
4 888##e f There are certain Sequences that you should know on sight:COMMON SEQUENCESe f## % ) "' $# '% 8: e f$$ * #( )" #%$ (#* 8: e f8" % * "' #& $' #: e f8" ) #( '% "#& #"' $: e f8x" # ' #% "#! (#! : The last of these is the sequence of , which you may not be familiar with. The thfactorials8term in this sequence (written , and pronounced factorial ) is the product of all the whole8x8numbers between and . For example:" 8&x " # $ % & "#!.EXAMPLE 2 Find formulas for the following Sequences : (a)$ * #( )" #%$# ' #% "#! (b) $ % $ & % ' & ( ' ) (c)"' #& $' %* '% SOLUTION(a) This is the sequence . $8x8(c) Let's compare with :+888 " # $%&' +$ % $ & % ' & ( ' ) 8 For the latter three terms, the coefficient is , and the number inside the square root is88 #.)
5 This formula also works for the first and second terms: $ " $% # %Therefore, this is the sequence . 8 8 #(d) Each of the terms in this sequence is a perfect square. Indeed:8"#$%& + "' % #& & $' ' %* ( '% ) 8#####The number being squared is always , so this is the sequence .8 $8 $ ab# Special Sequences Two types of Sequences that we will encounter repeatedly are and arithmetic sequencesgeometric Sequences . An is a sequence for which each term is a constant plus the previousarithmetic sequenceterm. For example, in the sequence& ) "" "% "( each term is obtained from the previous term by adding . This number is called the $$commondifference, since it can be obtained from subtracting any two consecutive terms.
6 The formula for an arithmetic sequence is always a linear function:ARITHMETIC SEQUENCESIf is an arithmetic sequence with common difference , thene f+.8+ 5 some constant .5 EXAMPLE 3 Find a formula for the sequence .ef& ) "" "% "( SOLUTIONS ince the common difference is , the formula for this arithmetic sequence must$have the form+ 5 $88where is some constant. Since is supposed to be , the constant must be . Therefore, this5+&5#"is the sequence .ef# $8 A is a sequence for which each term is a constant multiplied by thegeometric sequenceprevious term. For example, in the sequence' "# #% %) *' each term is exactly times the previous term. The number is called the , since##common ratioit can be obtained by taking the ratio of any two consecutive terms.
7 The formula for a geometric sequence is always an exponential function:GEOMETRIC SEQUENCESIf is a geometric sequence with common ratio , thene f+<8+ 5<88for some constant .5 EXAMPLE 4 Find a formula for the sequence .ef' "# #% %) *' SOLUTIONS ince the common ratio is , the formula for this geometric sequence must have the#form+ 5 #88where is some constant. Since is supposed to be , the constant must be . Therefore, this5+'5$"is the sequence .ef$ #5 EXAMPLE 5 Find formulas for the following Sequences : (a)* & " $ ( (b)"# ' $ $ $# %SOLUTION(a) Each term in this sequence is obtained by subtracting from the previous term. This is a%type of arithmetic sequence, with a common difference of . Therefore, thenegative%formula for this sequence must have the form+ 5 %88,where is some constant.
8 Since is supposed to be , the constant must be , so5+*5"$"this is the sequence .ef"$ %8(b) Each term in this sequence is half of the previous term. This is a type of geometricsequence, with a common ratio of . Therefore, the formula for the sequence must" #have the form+ 5 "5##888 ,where is some constant. Since is supposed to be , the constant must be , so5+"#5#%"this is the sequence .ef#% #8 The Limit of a Sequence You can take the limit of a sequence as as in the same way that you take the limite f+8 _8of a function as . The only difference is that there is one term for every positive0 B B _+a b8integer , while there is one value of for every real Ba bCONVERGENCE AND DIVERGENCEWe say the sequence if is a real f++888 _convergeslimIf is infinite or does not exist, the sequence.
9 Lim8 _8+divergesEXAMPLE 6 Does the sequence converge or diverge? 8#SOLUTIONS ince:lim8 _#8 _the sequence diverges to ._ EXAMPLE 7 Does the sequence converge or diverge? 8 "$8 %8 ###8 "_SOLUTIONWe have:limlim8 _8 _####8 "8"$8 %8 #$8$ Therefore, the sequence converges to ."$ EXAMPLE 8 Consider the sequence " " " " " " Though it may not be obvious, this sequence has a simple formula: the th term is .8 "a b8(You should check for yourself that this formula works.) As , the sequence will continue8 _to oscillate between and , and will therefore have no limit. " "(It approaches neither nor , " "so the limit does not exist. This is similar to the limit .) The sequence thereforelim sinB _Bdiverges. EXERCISES1 10 Find a formula for the general term of the+8sequence, assuming the pattern of the first few termscontinues.
10 1. " " " "$ * #( )" 2. " " " " "# % ' ) "! 3.$ % & '"' #& $' %* 4. " # ' #% "#!% * "' #& 5. ! $ ' * "# 6. $ # # " " ""## 7. " " " "$ & ( * 8. " # $ %"x $x &x (x "#$% 9. "! &! #&! "#&! 10. # " " " "$ $ ' "# #% 11 16 Determine whether the given sequenceconverges or diverges. If it converges, find the limit. 11. 12. + 8 "" $88 "8 #8## 13. 14. a b88" /8+ 88 "_8#arctan 15. 16. ef a b coscos# 88#118 "_)
