Section 3.1: Sequences and Series

Transcription

1 Section.: Sequences d Series Sequences Let s strt out with the definition of sequence: sequence: ordered list of numbers, often with definite pttern Recll tht in set, order doesn t mtter so this is one wy tht sequence differs from set. Also, repetition doesn t mtter in set but does in sequence: if number is repeted in sequence, it isn t considered duplicte d cnot be removed without chging the sequence. Sequences, like sets, c be finite or infinite. If sequence is finite, then either the lst term or the number of terms must be specified so tht it s cler where the sequence stops. Which of the following sequences re infinite? Which re finite? ) 7,, 5, 9, b), 4, 9, 6, 5, 6, 00 c) 4,,,,,,, b) d c) re finite, becuse their lst terms re given. ), however, goes on forever so is infinite. To begin with, let s exmine some sequences in detil. We will begin by looking for ptterns in ech sequence. Wht is the pttern for the following sequences? Wht is the next term for ech sequence? ) 7,, 5, 9, b), 4, 9, 6, 5, 6, 00 c) 4,,,,,,,

2 d), 6,, 4, e), 6, 5, 4, ) The pttern is tht you dd 4 to the previous term to get the next term. The next term is then. b) The pttern is tht if you sy tht is the first term d 4 is the second term, then n will be the nth term. So the next term fter 6 is 49. c) The pttern is to divide ech term by two (or multiply by hlf) to get the next term. So the term fter /6 will be /. d) The pttern is to multiply ech term by to get the next term. The next term is then 48. e) The pttern is to subtrct 9 from the previous term, so the next one is. Note tht in this previous exmple, the lst two sequences looked very similr for three of their first four terms. However, the third term is different so the pttern for the two sequences is not the sme d subsequent terms could look very different. Nottion We will use the nottion n for the nth term in sequence, where n is the index. For exmple, the first term would then be, the second term, d so on. The index n, then, is positive integer (or nturl number, if you like). Other nottions my strt their counting with o being the first term. For the purposes of this course, we ll stick to strting t n =. Defining Sequence There re three wys to define sequence: ) List ll of the terms, or enough terms to set up the pttern. If the sequence is finite, then either the lst term or the number of terms must be given. ) Give generl formul for the nth term. ) Give recursive formul for the nth term. Let s look t exmples of ech type. For instce, the sequences 7,, 5, 9, d, 4, 9, 6, 5, 6, 00 re exmples of sequences defined by listing the terms.

3 Generl Formul A generl formul is formul tht gives n s function of n only. Let s look t the following exmples to exmine some sequences defined in this wy. Give the first four terms of the sequence given by the generl formul 4n. n = 4n +, so = 4 + = 7 = 4 + = = 4 + = 5 4 = = 9 The first four terms re then 7,, 5, d 9. This is the sme sequence tht ws given s prt ) in the first two exmples of this section. n Give ll terms of the sequence given by the formul n for n5. This is finite sequence, since restrictions hve been plced on the vlues of n. The terms re then:

4 You c see from the previous exmples tht the generl formul llows you to clculte n for y vlue of n. The very useful thing bout the generl formul is tht you don t need to know the previous term to clculte prticulr term. For instce, if you wt to know the 50 th term of the sequence 7,, 5, 9,, you c determine tht the pttern is to dd 4 to the previous term to get the next term. However, to get the 50 th term, you d hve to clculte the 49 th first, but the 49 th requires the 48 th, d so on. But if you insted use the expression 4n, which gives the sme sequence, then the 50 th term is just 50 4n d there s no need to clculte preceding terms. Hdy! Recursive Definition A recursive formul gives formul for the next term in terms of the previous one. For exmple, in our old friend 7,, 5, 9,, the next term is found by dding 4 to the previous term: 4. However, tht s not enough informtion to uniquely define the series becuse you don t know where to strt. A complete definition must include the first term lso. Therefore, the recursive definition for our old friend 7,, 5, 9, would be 7 Recursive definitions, then, must specify the first term or terms d lso the rule which llows you to clculte the next term from the previous term or terms. 4 Clculte the first four terms of the sequence given by 0 The first term is lredy given,. Then

5 Give recursive formul for the sequence, 6, 8, 54, The pttern is tht the next term equls the previous term times three. Therefore, Recursive definitions hve the sme drwbck tht we ve seen before: if we wt to know the 00 th term, we need to clculte the 99 th first, d so on. Only the generl formul llows us to clculte ech term directly without knowing the previous one. Fiboncci sequence The Fiboncci sequence is the most fmous exmple of recursive sequence:,,,, 5, 8,, The pttern c be quite difficult to spot you get the next term from the sum of the two previous terms. The recursive formul for this sequence is therefore n n n Here, the first two terms must be given to strt off with so tht you re then ble to clculte the third term from the previous two. Series A series is the sum of sequence, which c be finite or infinite. Here re two exmples: ) b) Nottion The sum of the first n terms of sequence is denoted by S n (lso sometimes clled the nth prtil sum). If the series is finite, it could be the sum of ll of the terms. S is how we write the sum of infinite series, like the second exmple bove.

6 For the series , clculte S d S 4. S = = 60 S 4 = = 88 However, it s esy to see tht this method becomes very cumbersome for lrge vlues of n. We ll develop some more efficient methods in the next two sections. Sigm nottion It s esy to tke sequence in list form d trsform it into series by chging ll of the comms to + signs. However, wht if you re given the generl formul insted? For exmple, let s tke 7,, 5, 9, which we know to be 4n. Since the generl form is so useful for finding n when n is lrge, it would be nice if we could retin tht informtion while writing our sum. To do so, we ll introduce new nottion clled sigm nottion. It uses the Greek letter sigm (the uppercse one): Σ, which is commonly used to me sum of. Let s look t exmple of sigm nottion d discuss wht ll of the prts me. Consider the following 5 i (4i ) The letter i is index here, d it runs from the vlue given t the bottom of the sigm to the number t the top of the sigm in steps of. Here, i runs from to 5. We re summing, then, the vlue of 4i + for ech vlue of i s it runs from to 5: 5 i Let s look t more exmples. i i i i4 i5 (4i )

7 Clculte i (i 5) i i i i (i 5) Clculte 8 j 9 j6 9 j6 j 6 j 7 j 8 j 9 8 j Clculte 6 k 6 j k k k 4 k 5 k 6 5 The tricky thing bout the lst one is deciding how my terms there re. You my, s is shown bove, write out ll of the possible vlues of the index. Or you my use the following nifty rule:

8 # terms = lst first + For instce, the lst exmple hd the index running from to 6. The number of terms, then, for tht series is 6 + = 5. Write the following series in sigm nottion: Let s pick our index first. If we wt to be lzy, insted of strting our index t, we could strt t d our series would be 0 k Other cceptble swers would involve chging our strting point for the index 9 to give j or i or even l 55 j fvourite number. 8 i0 k 65 l57 if 57 hppens to be your Write the following sequence in sigm nottion: j To write infinite series in sigm nottion, you just replce the finl vlue of the index with. j