Sequences, Series and Convergence with the TI 92. Roger G. Brown Monash University


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1 Sequeces, Series ad Covergece with the TI 92. Roger G. Brow Moash Uiversity Itroductio. Studets erollig i calculus at Moash Uiversity, like may other calculus courses, are itroduced to sequeces ad series i the same sectio of work. These studets have difficulty distiguishig betwee a sequece ad a series. These difficulties are compouded by the eed for a sophisticated level of algebra as well as a soud kowledge of limits whe dealig with covergece of a sequece or a series. The difficulties have bee recogised by other authors icludig Morrel (992) The studets completig the first semester of the first year calculus course at Moash Uiversity, agai demostrated a poor uderstadig of the differece betwee a sequece ad a series as well as determiig whether a series was coverget or ot. I the secod semester, a group of studets icludig some who were repeatig the subject, were provided with a ivestigative approach to the topic by utilisig the multiple represetatios of the TI 92. The studets were ot oly able to graph ad tabulate sequeces, partial sums of series, they were also able to compare algebraically derived limits for both sequeces ad series. The ability to use the symbolic algebra properties of the TI 92 further assisted the studets i developig a uderstadig of the differece betwee a sequece ad a series. Studet Activities. A sample studet activity follows: The Geome Tree A demostratio usig Cabri o the TI 92 was used to costruct the sequece of Geome Trees. Studets commeced by coutig the umber of segmets. Most of the studets were able to determie the sequece for the legth of the ew segmets added, as well as write the formula for the th term. The studets were the asked to determie the partial sums for the legth of a path, from the base to the ed of a brach, before writig the series for the legth of a path. Fially they were also required to determie the partial sums ad the series for the umber of segmets.
2 Oce the studets had developed the sequece formula for the legth of the brach ad the series formula for the both the legth of the path ad the umber of braches they were the able to plot the graphs usig the sequece mode of the TI 92. A visual ispectio of the tables ad graphs allowed the studets to see whether the series coverged or diverged. The legths of the ew segmet added show both by the table ad the graph. l r = r 2 The partial sums for the legth of the path where represets the umber of braches are also show both by the table ad the graph. The formula for the partial sum is give by p = r r = 2 The partial sums for the umber of braches is also show, by both the table ad the graph. The formula for the partial sums is give by r b = 2 r = This ituitive itroductio allowed the studets to firstly develop a uderstadig of covergece ad divergece ad prepared them for a ivestigatio ivolvig the use of limits o the TI 92. By usig the split scree mode of the TI 92 they were able to view both the graph of a sequece (icludig a sequece of partial sums) ad a table of values for the terms of a sequece. The compariso of a umber 2
3 of examples eabled them to ituitively determie that the limit of the sequece of terms for a covergig series was 0, while for a diverget series ay value was possible icludig 0 ad. By comparig the graph of the partial sums of the legth of the path with the limit of the sequece it was see by the studets that the series is covergig ad that the limit of the correspodig sequece of terms is zero. The graph of the partial sums of the total umber of segmets portrays the series as diverget ad this is cofirmed by the limit of the correspodig sequece of terms which is ot equal to zero. The use of cotextual examples is importat as it provided studets with visual support to assist the developmet of their uderstadig of the terms sequece, partial sum ad series, as well as experiece with the summatio sig. Havig completed cotextual examples related to the Koch Sowflake ad the Sierpiska triagle they the moved oto examples ivolvig other covergig or divergig series such as the followig Example. Cosider the series. Plot the sequece of partial sums ad the ( + ) = determie whether the series coverges or ot. 3
4 Example 2. Cosider the series. Plot the sequece of partial sums ad the + = determie whether the series coverges or ot. I this example it was importat for the studets to recogise that the sequece cosistig of the terms of the series coverges to ad therefore the series is diverget. RESULTS At the coclusio of the sessios o Sequeces ad Series the studets were give a test coverig the work completed. A portio of the questios were the same as o the test i first semester, while others were adapted to suit the differet approach used i secod semester. Uiversity policy determied that the studets were oly allowed to use scietific calculators i the test. The results for the comparative questios are show i Table. 4
5 Questio. lim Determie whether x 0, exists or x 2 ot. Calculate the limit if it exists. Questio 2. Determie whether the followig sequece coverges or ot. If it coverges fid the value that it coverges to. = Questio 3. State whether the followig series diverges or coverges, givig reasos = First semester % correct = 43 23% ot applicable 23% Secod semester % correct = 28 8% 7% 54% Table. Comparative results from semester ad 2 for similar questios. Whe comparig the performaces from questios 2 ad 3 i semester two, oly studets or 39 % of the group were able to provide correct solutios to both questios, idicatig that studets were still havig difficulty whe comparig the covergece of a sequece ad that of a series. The results i the table demostrate that over half of those studets tested were able to recogise ad justify the correct solutio for questio 3 which is a marked icrease o the first semesters result. Studets were also required to costruct cocept maps usig at least the followig terms: sequece, series, partial sums, limits, covergece ad divergece. For may of the studets this was the first time they had bee asked to draw a cocept map. These cocept maps were scored accordig to the methods of Novak ad Gowi (984) ad the the scores from the cocept maps were compared with the studets resposes o the test. The studets' performace o the test was reflected i their costructio of the cocept maps. Studets lack of uderstadig of the differece betwee the covergece of a sequece ad that of a series was evidet. By cosiderig the cocept maps ad the test results it appeared that the studets' difficulty lay i the otio of fidig the limit as approached ifiity for the covergece of a sequece, i cotrast with the limit as approached ifiity for the terms of a series beig zero whe a series coverges. 5
6 Coclusio The effects of the work completed usig both a cotextual base ad the Texas TI 92 has ehaced the performace of studets i the first year calculus course at Moash Uiversity. Although the results from this alterative method of istructio are ecouragig, further research will eed to be udertake to determie the ogoig effectiveess of these methods. Particularly by developig the studets ability to recogise the uiqueess of the limit whe cosiderig the covergece or divergece of a series. Refereces Morrel, J. H. (992). A getle itroductio to ifiite series usig a graphig calculator. Primus 992 Vol 2 (). Novak, J.D. & Gowi, D. B. (984). Learig how to lear. Cambridge Uiversity Press: Cambridge. 6
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