Math 112 Fall 2018 Lab 8

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1 Ma Fall 08 Lab 8 Sums of Coverget Series I Itroductio Ofte e fuctios used i e scieces are defied as ifiite series Determiig e covergece or divergece of a series becomes importat ad it is helpful if e sum of a coverget series ca be determied Most of e tests taught i Chapter ca determie covergece but do ot allow us to fid e sum of e series Oe exceptio is a coverget geometric series which has e sum, where a is e first term of e series ad r is e commo ratio For ay series, if ere is a formula for e a s () r partial sum, s, e exact sum is s lim s () if e limit exists If a series is a telescopig series, we ca fid a formula for s ad use () to determie its covergece ad sum Series at coverge may or may ot have a kow sum, however For may series, we ca oly estimate e sum II Estimatig Sums Give a coverget series, ay Example : We kow e series coverges How would you support is coclusio? partial sum may be used to approximate e sum of e series

2 , is coverget ad oe possible approximatio of e sum 9 is s Of course, a better approximatio is s 9 8 ad s , is better still I fact, we kow e partial sum becomes a better approximatio as icreases Why? III Error Every approximatio is just at a approximatio raer a e exact value So every approximatio has a associated error I e previous example, e first four terms of e series are used to calculate s, is approximatio has a error equal to a a a7 Geerally, e error i usig s to approximate e sum of a coverget series is equal to e sum of all e remaiig terms That is, Error = actual sum approximate sum or Error = s - s This error is called e remaider which makes sese as it is made up of e remaiig terms ot used to calculate e approximatio, s The otatio R deotes e error i usig s to approximate s So, R s s a a a () If we caot fid e exact sum of a series, e we certaily caot fid e exact remaider associated wi a approximatio However, if e remaider ca be boud, we ca improve e approximatio of e sum If e Itegral Test ca be used to determie covergece of a series, it ca also be used to fid bouds for R The sketches below illustrate how R is boud betwee two area calculatios

3 So e error is boud as follows: x x f dx R f dx () The exact sum is somewhere betwee e approximatio plus e smallest error ad e approximatio plus e largest error That is, x x s f dx s s f dx () If a Alteratig Series is coverget, e error i approximatig e sum wi s ca also be boud The error is o greater a e absolute value of e first eglected term i s Example : The alteratig harmoic series show below is coverget 7 7 If s is used to approximate e sum of e series, we kow e remaider 0 is equal to e sum of all e remaiig terms Oce agai, we caot fid e exact remaider but we ca fid a upper boud for e remaider Notice what happes if we approximate e remaider, addig oe more term i each approximatio: R We see R R The error cotiues to get smaller ad smaller, as more terms are added Thus, e error (i absolute value) will be o greater a b Geerally, for series b e error is at satisfy e coditios of e Alteratig Series Test, e size of R s s b ()

4 Ma F8 Lab 8 Name: Sectio: Score: EXERCISES: For each series below, verify it is coverget State e ame of e series ad/or tests used to verify is ad iclude steps at justify your coclusio If e series coverges, fid e exact sum (a) k k (b)

5 (a) Show e series coverges by usig e Itegral Test Remember to show e l required coditios of e Itegral Test are met ad evaluate e itegral directly (b) Approximate e sum of e series usig s (calculator) (c) Fid e largest error expected i usig e approximatio i (b) for e sum

6 (a) Approximate e sum of e coverget series,usig s (calculator) (b) Fid e largest error expected i usig is approximatio for e sum (c) It ca be show at, alough e proof is beyod e scope of is course Suppose we kow, however, e formulas for e partial sums of two oer series such at Use e iequality above ad e Squeeze Theorem for sequeces to show

7 (a) Show e series coverges by usig e Alteratig Series Test (b) Approximate e sum of e series usig s (calculator) (c) Fid e largest error expected i usig is approximatio for e sum (calculator) L L terms L partial sums Helpful Calculator Stuff: We ca list values of e sequece of terms ad e sequece of partial sums for e series To clear y= ad lists: d, MEM,, eter Brig up table as show: STAT, ENTER Arrow over ad/or up to highlight L : d, LIST, OPS, seq(x,x,,00,) ENTER This displays e -values from to 00 i L Arrow right ad up to highlight L Eter e a formula, replacig wi L, e ENTER This displays e sequece of terms i e series Arrow right ad up to highlight L Eter: d, LIST, OPS,, ENTER, d, L, ENTER This displays e sequece of partial sums of e series a 7

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

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