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8.1 notes

# 8.1 notes - SECTION 8.1 SEQUENCES A sequence is an ordered...

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1 SECTION 8.1 SEQUENCES A sequence is an ordered list of numbers: a 1 , a 2 , a 3 , … Notation: Sequences may bedefined by giving a formula for the n ’th term : Ex. 1a. 0 1 2 n n =    a n = 1 2 n First four terms: Ex. 1b. 1 3 21 n n n = + a n = 3 n n + First four terms: Sequences may also be defined recursively Ex 1c. The Fibonacci sequence { f n }: f 1 = 1, f 2 = 1, f n = f n – 1 + f n – 2 for n 3. Write out the first 12 terms of the Fibonacci sequence Ex. 1c. Write out the first 6 terms of the sequence 0 sin 4 n n p = . Graphing Sequences using the TI-89/92 (TI-82, -83, calculators will do this too. For TI-86, see note on page 568.) Press MODE and change the graph mode to SEQUENCE , press ENTER twice. Press ( green ) Y= , and observe that the entries in this menu appear in pairs: u1, ui1,u2,ui2, etc. and that the variable for sequences is n . You enter the expression for the term in terms of n in the u’s. If your sequence is given recursively (the value of a term depends on previous terms), you’ll need to give the first term on the ui line. (It’s strange notation!) We’ll plot the first 20 terms from the sequences of example 1a and 1b. On the command line enter u1(n) = 1/2^n . It is not necessary to use ui1 since 1/2^n is an explicit formula. Next, enter u2(n) = 3*n/(2*n+1); again ui2 is not used. Press ( green ) WINDOW and set the values as follows: nmin = 0, nmax = 20, plotstrt = 1, plotstep = 1, xmin = 0, xmax = 20, xscl = 1, ymin = 0, ymax = 2, yscl = .2 Lastly, press ( green ) GRAPH and you’ll see the lower set of dots representing the numbers of the sequence 0 1 2 n n = approaching , and the upper set representing the sequence 3 n n + approaching the value . You can see the values by pressing F3 ( Trace ) and moving the cursor to the right or up and down between sequences. You CAN move the cursor to the left, but it is v-e-r-y slow!

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2 Another convenient way to see the values of the terms is by setting up a table. Set tblstart = 1, tbl = 1, and Independent: Auto. Press ENTER , then press ( green ) TABLE . Use your calculator to fill in the table below n 1 2 3 4 5 6 7 8 9 10 0 1 2 n n =    3 21 n n + For Maple instructions, see the worksheet series2002.mws . What happens to f n (the Fibonacci sequence) as n →∞ ? What happens to 0 sin 4 n n p = as n ? Definition of a the Limit of a Sequence A sequence { a n } has a limit L , and we write , or , if Here’s what can happen with sequences If lim n n aL = exists ( L is a real number), we say that the sequence… If lim n n a →∞ does not exist, we say that the sequence… If a n grows larger and larger as n increases, we write lim n n a =∞ and say the sequence… Use this terminology to describe examples 1a – d. 1a 1b 1c 1d Notice that taking the limit of a sequence as n is very like taking the limit of a function f ( x ) as x . What is the difference?
3 Theorem: If a n = f ( n ), and li m () x fx →∞ = L (exists), then li m li n nx a →∞ = = L .

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8.1 notes - SECTION 8.1 SEQUENCES A sequence is an ordered...

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