Summing Up Geometric Series

Transcription

1 Summig Up Geometric Series About the Lesso I this activity, studets will explore ifiite geometric series ad the partial sums of geometric series. The studets will determie the limits of these sequeces ad series usig tables ad graphs. As a result, studets will: Derive ad apply a formula for the sum of a ifiite coverget geometric series. Use the template to verify the formula for the sum of a ifiite series i specific cases. Prove ad apply the ratio (of cosecutive terms) test to prove a series coverget or diverget. Prove that a ecessary coditio that a geometric series coverges is that r < where r is the commo ratio. Vocabulary commo ratio ifiite series partial sum covergece divergece Teacher Preparatio ad Notes Studets should already be familiar with sequeces, partial sums, ad the defiitio of coverget ad diverget series. This activity is desiged to be teacher-led. You may use the followig pages to preset the material to the class ad ecourage discussio. Studets will follow alog with their calculators. Although the majority of the ideas ad cocepts are oly preseted i this documet, be sure to cover all the material ecessary for studets comprehesio. Before begiig the activity, studets should press y [mem] ad select 4:ClrAllLists to clear all data from their lists. Activity Materials Compatible TI Techologies: Tech Tips: This activity icludes scree captures take from the TI-84 Plus CE. It is also appropriate for use with the rest of the TI-84 Plus family. Slight variatios to these directios may be required if usig other calculator models. Watch for additioal Tech Tips throughout the activity for the specific techology you are usig. Access free tutorials at rs/pd/us/olie- Learig/Tutorials Ay required calculator files ca be distributed to studets via hadheld-to-hadheld trasfer. Lesso Files: Summig_Up_Geometric_Series _Studet.pdf Summig_Up_Geometric_Series _Studet.doc TI-84 Plus* TI-84 Plus Silver Editio* TI-84 Plus C Silver Editio TI-84 Plus CE * with the latest operatig system (2.55MP) featurig MathPrit TM fuctioality. 205 Texas Istrumets Icorporated educatio.ti.com

2 Summig Up Geometric Series Problem Ifiite Series A ifiite series ca be defied as = a + a 2 + a a +..., where a, a 2, ad a 3 are terms of the series. a = Studets begi by recogizig patter withi these series ad writig th term expressio for a give series.. Fid the ext three terms of each ifiite series. a b c Aswer: Aswer: Aswer: Studets should see for questio.c. that the first term, 2, is the same as a i a geometric series, meaig a + ar + ar ar + 2. Write a expressio i terms of that describes each of the above series usig sigma otatio. Aswers: a. or b. c Problem 2 Fidig the Sum of a Geometric Series Studets ca fid the partial sum of a geometric series. I this problem, studets will fid a partial sum of two geometric series. Press ƒ _ Á to select 2:summatio Σ( Use the arrow keys to maeuver. Fid the partial sum of these geometric series. To fid the sum of a series, press ƒ _ Á for summatio. Use the arrow keys to maeuver. Notice that you eed to type aother set of paretheses withi the paretheses that are supplied. To show 205 Texas Istrumets Icorporated 2 educatio.ti.com

3 Summig Up Geometric Series the decimal, press» Á Í. 3a. 8 3 = Aswer: b. 6 = Aswer: = = Aswer: Problem 3 Covergece ad Divergece of Geometric Series A geometric series with first term a ad commo ratio r is give by ar = a + ar + ar ar +..., a 0. A geometric series diverges if r. It coverges to the sum ar = a r if 0 < r <. These coditios must be used i determiig whether a series diverges or coverges. It is also worth otig that a series may diverge, but will ot ecessarily diverge to ifiity. A value r that is less tha will result i a series that diverges ad has terms whose sigs alterate from positive to egative, ot divergig to ifiity. Aother importat ote to studets is that a series coverges or diverges if the sequece of the partial sums coverges to its sum or diverges. 205 Texas Istrumets Icorporated 3 educatio.ti.com

4 Summig Up Geometric Series Studets will eter the values of a sequece i a list to further ivestigate the behavior of the geometric series. Press Í to access the table of data scree. I L, eter seq(x,x,,50) i the top most cell. The seq( commad ca be foud by pressig y [list] ad arrowig over to OPS ad selectig 5:seq(. Eter the iformatio i the seq meu exactly as show i the scree to the right. I the top most cell of L2, type L ad press Í Texas Istrumets Icorporated 4 educatio.ti.com

5 Summig Up Geometric Series Studets will ow graph the series by geeratig a list with the cumulative sums of the terms of the sequece. To do this, move to the top most cell of L3, press Í, the press y [list] ad arrow over to OPS ad select 6:cumSum(. The type y Á e ad press Í. The first 50 partial sums of the series L will be displayed i = Next, we ca view a graph for each series by creatig a scatter plot of the values of the partial sums of the series. To create a scatter plot, select y o [stat plot] À. Set up as show i the figure to the right. To view the graph, press q 9:ZoomStat. Pressig p ad chagig each of the followig: Xscl: 2 Yscl: 0.2 It will give studets a better view of the graph of the partial sums if grid lies are tured o. Tech Tip: If your studets are usig the TI-84 Plus CE have them tur o the GridLie by pressig y q [format] to chage the graph settigs. If your studets are usig TI-84 Plus, they could use GridDot. 205 Texas Istrumets Icorporated 5 educatio.ti.com

6 Summig Up Geometric Series To fid the sum of a geometric series, studets must take the limit of the th sum. The series is a special case of the geometric series. Oe way of attemptig this problem is to list the partial sums of the series the determie the th sum. The partial sums for this series would be S = 2, S 2 = 3 4, S 3 = 7 8, S 4 = 5 6, Guide studets so they see that the deomiator is 2 raised to the th power ad the umerator is always less tha the deomiator. This gives 2 2. Take the limit of the sum lim 2 2. By the geometric series test, the series coverges because 0 < 2 <, ad the sum is, by fidig the limit. Aother way is to use the geometric series test that ca be stated ar = a or ar = a. It is importat to ote to r r = 0 studets the idex of the series. The three graphs to the right represet the series. Notice that, whe traced, the values of the series approach. 205 Texas Istrumets Icorporated 6 educatio.ti.com

7 Summig Up Geometric Series Determie the covergece or divergece of each of the followig series. Create a scatter plot of the values or the partial sums to aid i determiig the behavior of each series. 5. Aswer: This series coverges to. Repeatig the same steps for 6 ad 7 will yield the followig results Aswer: This series coverges to Aswer: The series diverges. Have studets sketch the graphs i the space provided o their worksheet. Optioal: Ecourage studets to participate i a class discussio. 205 Texas Istrumets Icorporated 7 educatio.ti.com