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{"sheet":"Approximating Pi","index":0,"n_rows":1520,"n_cols":11,"rows_kept":1520,"cut":false,"hidden":false,"merged":[[24,7,25,11],[29,7,30,11],[1,1,1,11],[4,1,4,2],[17,8,17,10],[19,8,19,9],[18,8,18,9],[7,1,7,5],[12,1,12,5],[3,5,5,6],[15,2,15,5]]}
[]
[null,"You can use an excel table, the Lebiniz formula, and the alternating series estimation theorem to correctly compute as many of the digits of π as you want!"]
[]
[null,null,null,null,null,"          For more details see page 774 in the Stewart text, or\n                  en.wikipedia.org/wiki/Leibniz_formula_for_π"]
[null,"Lebiniz formula: "]
[]
[]
[null,"Using the Table (below), how many terms do you need to ensure that you have …"]
[null,null,"1. One digit of accuracy (π starts with a 3)?"]
[null,null,"2. Two digits of accuracy (π starts with 3.1)?"]
[null,null,"2. Two digits of accuracy (π starts with 3.14)?"]
[]
[null,"Using the Macro (the button and box on the right), "]
[null,null,"1. Fill in the table on the next spreadsheet."]
[null,null,"2. Decide if it is wise to try to use this method to find π to 100 decimal places!"]
[null,null,"3. Read en.wikipedia.org/wiki/Leibniz_formula_for_π to learn more!"]
[]
[null,null,"n","n^th term (a_n)","Current approximation of π (4*s_n)","Alternating Series bound on the error \nof the previous approximation of π (4*|error of s_{n-1}| ≤ 4*a_n)",null,null,"You can compute more digits of π using a computer program.  (Like this Excel Macro!)"]
[null,null,0,{"$f":"(-1)^C19/(2*C19+1)","v":1},{"$f":"4*SUM($D$19:D19)","v":4},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":4},null,null,"Desired # of digits of π:",null,3]
[null,null,{"$f":"C19+1","v":1},{"$f":"(-1)^C20/(2*C20+1)","v":-0.33333333333333331},{"$f":"4*SUM($D$19:D20)","v":2.666666666666667},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":-1.3333333333333333},null,null,"Approximation of π:",null,3.1431875489073047]
[null,null,2,{"$f":"(-1)^C21/(2*C21+1)","v":0.2},{"$f":"4*SUM($D$19:D21)","v":3.4666666666666668},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":0.8},null,null,"Terms needed:",null,627]
[null,null,3,-0.14285714285714285,{"$f":"4*SUM($D$19:D22)","v":2.8952380952380956},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":-0.5714285714285714}]
[null,null,4,0.1111111111111111,{"$f":"4*SUM($D$19:D23)","v":3.3396825396825403},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":0.44444444444444442}]
[null,null,5,-9.0909090909090912E-2,{"$f":"4*SUM($D$19:D24)","v":2.9760461760461765},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":-0.36363636363636365},null,"WARNING!"]
[null,null,6,7.6923076923076927E-2,{"$f":"4*SUM($D$19:D25)","v":3.2837384837384844},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":0.30769230769230771},null,"Be sure to save all open documents before playing with \"Desired # of digits\".  "]
[null,null,7,-6.6666666666666666E-2,{"$f":"4*SUM($D$19:D26)","v":3.0170718170718178},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":-0.26666666666666666}]
[null,null,8,5.8823529411764705E-2,{"$f":"4*SUM($D$19:D27)","v":3.2523659347188767},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":0.23529411764705882}]
[null,null,9,-5.2631578947368418E-2,{"$f":"4*SUM($D$19:D28)","v":3.0418396189294032},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":-0.21052631578947367}]
[null,null,10,4.7619047619047616E-2,{"$f":"4*SUM($D$19:D29)","v":3.2323158094055939},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":0.19047619047619047}]
[null,null,11,-4.3478260869565216E-2,{"$f":"4*SUM($D$19:D30)","v":3.0584027659273332},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":-0.17391304347826086},null,"Computing π is time consuming,                                      and might crash Microsoft Excel!"]
[null,null,12,0.04,{"$f":"4*SUM($D$19:D31)","v":3.2184027659273333},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":0.16}]
[null,null,13,-3.7037037037037035E-2,{"$f":"4*SUM($D$19:D32)","v":3.0702546177791854},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":-0.14814814814814814}]
[null,null,14,3.4482758620689655E-2,{"$f":"4*SUM($D$19:D33)","v":3.2081856522619439},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":0.13793103448275862}]
[null,null,15,-3.2258064516129031E-2,{"$f":"4*SUM($D$19:D34)","v":3.0791533941974278},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":-0.12903225806451613}]
[null,null,16,3.0303030303030304E-2,{"$f":"4*SUM($D$19:D35)","v":3.2003655154095489},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":0.12121212121212122}]
[null,null,17,-2.8571428571428571E-2,{"$f":"4*SUM($D$19:D36)","v":3.0860798011238346},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":-0.11428571428571428}]
[null,null,18,2.7027027027027029E-2,{"$f":"4*SUM($D$19:D37)","v":3.1941879092319425},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":0.10810810810810811}]
[null,null,19,-2.564102564102564E-2,{"$f":"4*SUM($D$19:D38)","v":3.0916238066678399},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":-0.10256410256410256}]
[null,null,20,2.4390243902439025E-2,{"$f":"4*SUM($D$19:D39)","v":3.1891847822775961},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":9.7560975609756101E-2}]
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[null,null,22,2.2222222222222223E-2,{"$f":"4*SUM($D$19:D41)","v":3.1850504153525314},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":8.8888888888888892E-2}]
[null,null,23,-2.1276595744680851E-2,{"$f":"4*SUM($D$19:D42)","v":3.0999440323738079},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":-8.5106382978723402E-2}]
[null,null,24,2.0408163265306121E-2,{"$f":"4*SUM($D$19:D43)","v":3.1815766854350325},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":8.1632653061224483E-2}]
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[null,null,29,-1.6949152542372881E-2,{"$f":"4*SUM($D$19:D48)","v":3.1082685666989471},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":-6.7796610169491525E-2}]
[null,null,30,1.6393442622950821E-2,{"$f":"4*SUM($D$19:D49)","v":3.1738423371907505},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":6.5573770491803282E-2}]
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[null,null,32,1.5384615384615385E-2,{"$f":"4*SUM($D$19:D51)","v":3.1718887352371485},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":6.1538461538461542E-2}]
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[null,null,34,1.4492753623188406E-2,{"$f":"4*SUM($D$19:D53)","v":3.1701582571925884},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":5.7971014492753624E-2}]
[null,null,35,-1.4084507042253521E-2,{"$f":"4*SUM($D$19:D54)","v":3.1138202290235744},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":-5.6338028169014086E-2}]
[null,null,36,1.3698630136986301E-2,{"$f":"4*SUM($D$19:D55)","v":3.1686147495715193},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":5.4794520547945202E-2}]
[null,null,37,-1.3333333333333334E-2,{"$f":"4*SUM($D$19:D56)","v":3.1152814162381861},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":-5.3333333333333337E-2}]
[null,null,38,1.2987012987012988E-2,{"$f":"4*SUM($D$19:D57)","v":3.1672294681862381},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":5.1948051948051951E-2}]
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[null,null,41,-1.2048192771084338E-2,{"$f":"4*SUM($D$19:D60)","v":3.1177865017588782},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":-4.8192771084337352E-2}]
[null,null,42,1.1764705882352941E-2,{"$f":"4*SUM($D$19:D61)","v":3.1648453252882898},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":4.7058823529411764E-2}]
[null,null,43,-1.1494252873563218E-2,{"$f":"4*SUM($D$19:D62)","v":3.118868313794037},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":-4.5977011494252873E-2}]
[null,null,44,1.1235955056179775E-2,{"$f":"4*SUM($D$19:D63)","v":3.1638121340187562},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":4.49438202247191E-2}]
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[null,null,47,-1.0526315789473684E-2,{"$f":"4*SUM($D$19:D66)","v":3.1207615795929895},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":-4.2105263157894736E-2}]
[null,null,48,{"$f":"(-1)^C67/(2*C67+1)","v":1.0309278350515464E-2},{"$f":"4*SUM($D$19:D67)","v":3.1619986929950512},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":4.1237113402061855E-2}]
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[null,null,50,9.9009900990099011E-3,{"$f":"4*SUM($D$19:D69)","v":3.1611986129870506},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":3.9603960396039604E-2}]
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[null,null,56,8.8495575221238937E-3,{"$f":"4*SUM($D$19:D75)","v":3.1591351638147658},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":3.5398230088495575E-2}]
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[null,null,58,8.5470085470085479E-3,{"$f":"4*SUM($D$19:D77)","v":3.1585405893071479},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":3.4188034188034191E-2}]
[null,null,59,-8.4033613445378148E-3,{"$f":"4*SUM($D$19:D78)","v":3.1249271439289967},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":-3.3613445378151259E-2}]
[null,null,60,8.2644628099173556E-3,{"$f":"4*SUM($D$19:D79)","v":3.157984995168666},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":3.3057851239669422E-2}]
[null,null,61,-8.130081300813009E-3,{"$f":"4*SUM($D$19:D80)","v":3.1254646699654138},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":-3.2520325203252036E-2}]
[null,null,62,8.0000000000000002E-3,{"$f":"4*SUM($D$19:D81)","v":3.1574646699654139},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":3.2000000000000001E-2}]
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[null,null,67,-7.4074074074074077E-3,{"$f":"4*SUM($D$19:D86)","v":3.1268875661065292},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":-2.9629629629629631E-2}]
[null,null,68,7.2992700729927005E-3,{"$f":"4*SUM($D$19:D87)","v":3.1560846463984999},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":2.9197080291970802E-2}]
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[null,null,70,7.0921985815602835E-3,{"$f":"4*SUM($D$19:D89)","v":3.1556764623074751},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":2.8368794326241134E-2}]
[null,null,71,-6.993006993006993E-3,{"$f":"4*SUM($D$19:D90)","v":3.1277044343354472},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":-2.7972027972027972E-2}]
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[null,null,73,-6.8027210884353739E-3,{"$f":"4*SUM($D$19:D92)","v":3.1280797568782575},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":-2.7210884353741496E-2}]
[null,null,74,6.7114093959731542E-3,{"$f":"4*SUM($D$19:D93)","v":3.15492539446215},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":2.6845637583892617E-2}]
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[null,null,76,6.5359477124183009E-3,{"$f":"4*SUM($D$19:D95)","v":3.1545791190866574},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":2.6143790849673203E-2}]
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[null,null,79,-6.2893081761006293E-3,{"$f":"4*SUM($D$19:D98)","v":3.1290931417757211},{"$f":"4*Table1[[#This Row],[n^th term (a_n)]]","v":-2.5157232704402517E-2}]
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{"sheet":"Graphing Digits vs # of terms","index":1,"n_rows":12,"n_cols":3,"rows_kept":12,"cut":false,"hidden":false,"merged":[]}
[]
[]
[null,"Decimal Digits ","Number of terms"]
[null,1]
[null,2]
[null,3]
[null,4]
[null,5]
[null,6]
[null,7]
[null,8]
[null,9]