Sequences and Series Using the TI-89 Calculator

Transcription

1 RIT Calculator Site Sequeces ad Series Usig the TI-89 Calculator Norecursively Defied Sequeces A orecursively defied sequece is oe i which the formula for the terms of the sequece is give explicitly. For example, a = is the defiig relatio for a + 1 o-recursively defied sequece, while a = a-1, a1 = 1 is a example of a recursively defied sequece. Terms of a orecursively defied sequece ca be geerated usig the built-i seq fuctio of the TI-89. Select 3:List, 1:seq from the Math meu (Fig. 1), obtaied by pressig the Math (= 2d 5 ) key. Fig. 1 Math 3:List Meu Optio Fig. 2 Math 3:List 1:seq Meu Optio To create the first 10 elemets of the sequece defied by the formula a = + 1, eter the expressio show i Fig. 3. This yields the sequece { , 10 3, 4, K, 11 } as show i Fig. 4. Fig. 3 Sytax of the seq Fuctio stoppig value for idex idex icremet seq(/(+1),,1,10,1) defiig formula startig value for idex Fig. 4 Geeratig terms of a sequece To calculate lim press the Math (= 2d 5 ) key ad choose A:Calculus, 3:limit Æ + 1 ad eter the expressio show i Fig. 5. Ifiity,, is the fuctio of.the limit of 1 is displayed o the home scree (Fig. 6).

2 RIT Calculator Site Sequeces ad Series Usig the TI-89 Calculator 2 Fig. 5 Sytax of the Limit Operator Fig. 6 The Limit of a Sequece value at which to calculate the limit limit(/(+1),, ) defiig formula variable Plottig the Terms of a Sequece The TI-89 has a built-i sequece plottig mode. To access it press the MODE key ad set Graph to SEQUENCE (Fig. 7). The Y= scree chages to reflect the ew mode (Fig. 8). Formulas for sequeces are etered i the slots labeled u1, u2, etc. (The slots labeled ui1, ui2, etc. are oly relevat for recursively defied sequeces. See below.) Fig. 7 Settig Graph Mode to Sequece Fig. 8 Y= Widow i Sequece Mode I eter the sequece defied by the formula a = i u1. I select the umber of terms + 1 to plot by eterig the appropriate settigs i the Widow scree (press F2 ). The mi ad max settigs show i Fig. 10 will geerate the first 30 terms of the sequece. The xmi ad xmax settigs cotrol how much of the sequece is displayed. Fig. 9 Eter the Sequece Formula Fig. 10 Widow Settigs Formula Igore this for orecursive sequeces Sets the umber of terms that are geerated Cotrols what s displayed

3 RIT Calculator Site Sequeces ad Series Usig the TI-89 Calculator 3 Fig. 11 Sequece Plot Fig. 12 Table SetUp Fig. 11 shows the sequece plot. The umerical values of the terms of the sequece ca be ispected usig the Table features of the calculator. Press TblSet (= F4 ) to brig up the TblSet widow (Fig. 12). The value of tblstart should agree with the Widow variable mi. Press TABLE (= F5 ) to display the table (Fig. 13). Sice sequeces defied i the Y= widow are fuctios, they ca be evaluated at the Home scree (Fig. 14). Fig. 13 Sequece Table Fig. 14 Sequeces as Fuctios Recursively Defied Sequeces Recursively defied sequeces ca be defied i the Y= editor whe Graph mode 1 is set to sequece (Fig. 7). I Fig. 15 I defie the sequece u =, u1 = 2. Note that 3 - u -1 the iitial value is part of the defiitio of the sequece. Usig the widow settigs i Fig. 16 I obtai a graph ad a table (Figs. 17 ad 18). Fig. 15 Defiig a Sequece Recursively Fig. 16 Widow Settigs

4 RIT Calculator Site Sequeces ad Series Usig the TI-89 Calculator 4 Fig. 17 Plot of Sequece of Fig. 15 Fig. 18 Table for Sequece of Fig. 15 The TI-89 has may other sequece capabilities. For example, cobweb plots of recursively defied sequeces ca be costructed to study the log-term dyamics of a sequece. See the chapter of your maual etitled Sequece Graphig for more iformatio. Ifiite Series Partial sums of ifiite series ca be Fig. 19 Fidig a Partial Sum of a Series calculated o the Home scree by usig the seq ad sum fuctios together. (Both of these fuctios are foud i the List submeu of the Math meu. See Figs. 1 ad 2.) To fid the sum of the first 10 terms of the sequece a =, eter the expressio + 1 sum(seq(/(+1),,1,10,1)) as show i Fig. 19. If Mode is set to Auto or Exact, the aswer ca still be displayed i approximate mode by pressig ENTER. The sequece of partial sums of a ifiite series ca be costructed i the Data/Matrix Editor. Press the APPS key ad choosig 6:Data/Matrix Editor (Fig. 20) ad the 3:New (Fig. 21). Fig. 20 Data Editor Meu Optio Fig. 21 Creatig a New Data Set I the dialog box that follows, eter a ame i the Variable box ad press ENTER twice. This brigs up the Editor scree (Fig. 22).

5 RIT Calculator Site Sequeces ad Series Usig the TI-89 Calculator 5 Suppose we wish to fid the partial sums of the series 1 Â. Move the cursor to the 2 header cell c1 so that the cell is highlighted ad press ENTER. O the etry lie use the seq fuctio, obtaied by pressig the Math (= 2d 5 ) key ad the selectig 3:List, 1:seq, to create the formula to geerate the sequece of terms of the series. I Fig. 22 I eter seq(1/^2,,1,10,1) to geerate the first 20 terms of a = 1. Press ENTER to geerate the 2 sequece. Fig. 22 Geeratig a Sequece i the Data/Matrix Editor i=1 i Fig. 23 Computig the Partial Sums of a Series i the Data/Matrix Editor Move to the header cell labeled c2 ad press ENTER to access the etry lie. Locate the cumsum fuctio, obtaied by pressig the Math (= 2d 5 ) key, ad the selectig 3:List, 7:cumSum. Iside the paretheses, type c1 for the argumet ad the press ENTER. CumSum is the cumulative sum fuctio ad it geerates the partial sums of the series (Fig. 23). To see approximate values for the sums, chage the Mode settigs (Fig. 24) by pressig MODE ad chagig the Exact/Approx settig to Approximate. The data table will ow display umerical approximatios of the partial sums (Fig. 25). Fig. 24 Settig Approximate Mode Fig. 25 Approximate Values Displayed